![Finite-size effect of the excess FE at the melting state (ρσ 3 = 1.0409). Lines for the EC, IH, and SS references were computed by adding the FE of the corresponding size-dependent reference to the (sizedependent) FE differences computed via TI on the PR path. Results by Polson et al.[56] give similar results for the TI+FL+EC method. Lines labeled HybridInf are computed using a size-dependent TI+PR integral with the FE for an infinite-system reference.](https://www.researchgate.net/profile/Sabry-Moustafa/publication/299445192/figure/fig4/AS:614173033644041@1523441670803/Finite-size-effect-of-the-excess-FE-at-the-melting-state-rs-3-10409-Lines-for-the_Q320.jpg)
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HIGHLY EFFICIENT MOLECULAR SIMULATION METHODS FOR EVALUATION OF
THERMODYNAMIC PROPERTIES OF CRYSTALLINE PHASES
by
Sabry Gad Al-Hak Mohammad Moustafa
August 12, 2015
A dissertation submitted to the
Faculty of the Graduate School of
the University at Buffalo, State University of New York
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
Department of Chemical and Biological Engineering
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ProQuest Number: 3725965
Major advisor: Prof. David A. Kofke.
Committee members: Prof. Eva Zurek, Prof. Carl R.F. Lund, and Dr. Andrew J. Schultz.
ii
Acknowledgments
Freedom, trust, and respect. These are the three words that pretty well summarize Prof. Kofke’s relation
with his students. The space of freedom I lived in is extraordinary; progress of research depends, to large extent,
on students’ enthusiasm and hard work, and not on him forcing or (micro!) monitoring us along the way. The
freedom is one of the manifestations of of the amount of trust he puts in his students. Above all, respect is the
right word when it comes to communications, a behavior that never broken. I am not exaggerating if I say that
all this started from day one! This environment motivated me for hard working and made me love what I am
doing.
The multidisciplinary fields he knows allowed me to work on different problems throughout the years.
In addition, facilitating me to cooperate/interact with other research groups made it even more beneficial and
exciting. He is not a formal advisor when it comes to research; for example, coming on weekends is not a
burden for him if we have new ideas to explore. This made research a joy and ”fun” for us. This all is due to
his leadership abilities. I am really so grateful to you, Dr. Kofke; Thank you!
Second, I would like to express my deep gratitude to Dr. Andrew Schultz, a researcher who knows how to
get things done! Although he is a research scientist, his office is just side-by-side with graduate students! He is
always reachable and he is always ready (and likes) to help. Interestingly enough is that he does not, usually,
give answers; but rather a way of (scientific) thinking. In addition, things must be always done in the right way;
even if it took months! Thanks Dr. Andrew for your constant support and guidance.
I also thank Prof. Eva Zurek (chemistry department) for giving me the opportunity to interact with her and
with her students from whom I learned so much. Although she belongs to a different department, I have never
felt “strange” during my staying with her group. Despite the fact that it took long time exploring different
problems/systems, it ended up with one of the most exciting problem in the scientific field, the Earth’s inner-
core crystal structure. So, thank you Dr. Eva for your patience and understanding; I really appreciate it.
Also, I would like to thank my committee member Prof. Carl Lund. He is one of the few people I have
even seen who has an extraordinary teaching skills; this made me see the field from a broader perspective. I
also thank him for proofreading my dissertation and remarkable comments about my work.
iii
Last but not least, I would like to express my deep grateful to the efforts the graduate school made to make
this work possible. From a person studied on different campuses, I see UB to be a great place to conduct
research. In addition, the friendly group I worked in had a great impact on my progress; many thanks to: Jung
Ho Yang, Shu Yang, Ramachandran Subramanian, Weisong Lin, and Nate Barlow.
Above all, I thank God who always guides me to the right path that is always the most relevant thing to me,
even if it does not seem so at the beginning! I thank my mother for he endless support and love. Thanks are
due to my dad who built in me the dedication and hard work. I am so grateful to my wife, Nadia, who suffered
a lot during this journey, without much complains, but rather with extra support and love.
Contents
Acknowledgments iii
Abstract xiii
Publications xvi
1 Introduction 1
1.1 Molecular Simulation (MS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Free Energy Calculations of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Formalism and Methods 10
2.1 Free Energy and Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Free Energy Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Thermodynamic Integration (TI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Free-Energy Perturbation (FEP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Einstein Crystal (EC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Interacting Harmonic Crystal (IH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 System-based harmonic approximation: Lattice Dynamics (LD) . . . . . . . . . . . . 17
2.3.4 Self-consistent phonon approximation (SCPA) . . . . . . . . . . . . . . . . . . . . . 18
2.3.5 Soft-Sphere Crystal (SS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Thermodynamic Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Physical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Non-physical Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Material Properties from Free-Energy Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Free Energy of Hard Spheres Crystal: A Comparative Study 24
3.1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 Reference Temperature for PR Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Free Energy Results for Finite-Size Systems . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.3 Finite-Size Effects: Thermodynamic Limit Results . . . . . . . . . . . . . . . . . . . 31
3.3.4 Computation Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Free Energy of Clathrate Hydrates: Harmonic Approximation 38
4.1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Theory and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Lattice Dynamics of Rigid Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.2 Proton-Disorder Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.1 Structures generation and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.2 H-isomer averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.3 Finite-size effects: thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . 51
vi
4.4.4 Free-energy results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4.5 Phonon density-of-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4.6 Accuracy of the local-harmonic approximation . . . . . . . . . . . . . . . . . . . . . 56
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5 Direct Measurement of Anharmonic Properties of Solids 60
5.1 Introduction and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Harmonically-Mapped Averaging (HMA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.2 Application: Tand Vfree-energy derivatives . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Application to LJ potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.2 Difficulty of Measuring Free-energy Derivatives . . . . . . . . . . . . . . . . . . . . 67
5.3.3 Difficulty of Measuring Absolute Free-Energy . . . . . . . . . . . . . . . . . . . . . 74
5.3.4 Thermodynamic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.5 Effect of Mapping on the System Trajectory . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Application to embedded-atom model (EAM) . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Thermodynamic Stability of Iron Polymorphs in the Earth’s Inner-Core 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.1 The Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.2 Density-Functional Theory (DFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.1.3 Solving KS equations in periodic systems . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.4 Finite-temperature DFT: Mermin’s approach . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Crystal Structure of Earth’s Inner Core (EIC) . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
vii
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7 General Conclusions and Future Work 100
A Atomic-to-molecular force-constants transformation 103
B Atomic Hessian matrix of TIP4P water model 106
C Reduction theorem of the full system Hessian matrix 109
D Equivalence of different FE formulas 114
Bibliography 118
viii
List of Figures
1.1 Schematic diagram of multiscale simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3.1 MSD vs. Tfor different reference systems (EC, IH, and SS). . . . . . . . . . . . . . . . . . 29
3.2 Integrand of Eq. 2.5, ∂A/∂λ, for the SS reference and the PR path at density ρσ3= 1.0409. . 31
3.3 Finite-size effect of the FE difference between SS and HS systems at the melting condition. . . 32
3.4 Finite-size effect of the excess FE at the melting state. . . . . . . . . . . . . . . . . . . . . . . 33
4.1 The potential energy versus the net dipole moment of the proton-disordered sI, sII, and sH
hydrate structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 The 1×1×1sI harmonic free-energy using various approximations expressed as a difference
from the full exponential average. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Finite-size effects on the classical harmonic free energy for the sI, sII, and sH structures. . . . 53
4.4 Classical harmonic Helmholtz free energy for the sI, sII, and sH structures versus temperature. 54
4.5 Quantum harmonic Helmholtz free energy for the sI, sII, and sH structures versus temperature. 55
4.6 Normalized phonon density-of-states (PDOS) for sI, sII, and sH structures. . . . . . . . . . . . 56
4.7 Relative stability based on classical free energies of sI, sII, and sH structures using full and
local harmonic (LHA) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1 Conventional and HMA potential energy trajectories of LJ system. . . . . . . . . . . . . . . . 69
ix
5.2 Conventional and HMA estimations of anharmonic energy along two isochores of LJ system. . 70
5.3 Isochoric anharmonic heat capacity along unity isochore of LJ system. . . . . . . . . . . . . . 70
5.4 Difficulty ratio of different thermodynamic properties along unity isochore of LJ system. . . . 71
5.5 Difficulty ratio of different thermodynamic properties along unity isotherm of LJ system. . . . 71
5.6 Difficulty ratio of different thermodynamic properties along the coexistence line of LJ system. 72
5.7 Comparison of the uncertainties from the conventional and HMA measurement of Cv“direct”
and fitting approaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.8 Difficulty in measuring the free energy against standard methods. . . . . . . . . . . . . . . . . 73
5.9 (Color online) Finite-size effects of measuring the pressure directly (no correction) and using
the HarmFSC ≡Pharm(∞)−Pharm(N)correction. The corrected pressure is obtained by
adding this correction to the pressure from simulation at some N. Inset: Effect of temperature
on the finite-size effects of the corrected pressure. Lines are linear fits weighted by uncertainties. 75
5.10 (Color online) Long-range effects of measuring the pressure using three corrections: the stan-
dard LRC (assuming homogeneous medium), lattice LatLRC ≡Plat (∞)−Plat(rc), and har-
monic HarmLRC ≡Pharm(∞)−Pharm (rc). The corrected pressure is obtained by adding the
correction to the pressure from simulation at some rc. Note that some points from LRC and
LatLRC with short rcare out of the scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.11 (Color online) Normalized autocorrelation function of the conventional and mapped measure-
ments of the energy. A trajectory of 2×105samples is used. . . . . . . . . . . . . . . . . . . 77
5.12 (Color online) Number of independent energy samples out of 2×105measured samples. The
number of independent samples is estimated as Var/σ2with Var and σthe sample variance
and uncertainty, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.13 (Color online) Effect of mapping on the equilibaration rate of conventional and harmonically-
mapped averagings. All data are normalized to the initial energy U0(off the average Uavg). The
dotted lines are the normalized uncertainties from a 104MC production run. . . . . . . . . . . 79
5.14 (Color online) Convergence of the average energy with respect to MD time-step size. Red line
simply joins the points. All points are generated using the same integration time, t= 5000 (LJ
units). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
x
5.15 Pressure and energy uncertainty ratios of EAM model along v = 7 ˚
A3/atom isochore. . . . . . 81
5.16 Pressure and energy uncertainty ratios of EAM model along T= 3000 K isotherm. . . . . . . 82
5.17 Conventional and HMA estimations of the specific anharmonic energy of BCC structure of iron
along v = 8.33 ˚
A3/atom isochore. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.18 Effect of MD time step (dt) on the accuracy of measuring EAM anharmonic energy using
conventional and HMA methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1 Flowchart of solving KS equation iteratively. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Pressure and energy uncertainty ratios of DFT model along v = 6.35 ˚
A3/atom isochore of DFT
model................................................ 94
6.3 Pressure and energy uncertainty ratios of DFT model along T= 6000 K isotherm. . . . . . . 95
6.4 Anharmonic energy along v = 6.35 ˚
A3/atom isochore of DFT model. Red line is a second-
order polynomial fit of the HMA data. The fitting parameters (in increasing polynomial powers)
are a0= 1.35(7) ×10−9eV/K2,a1= 3(5) ×10−14 eV/K3, and c2= 2.6(6) ×10−17 eV/K4. 96
6.5 Conventional and HMA measurements of pressure along T= 6000 K isotherm of DFT model.
The line is a second order polynomial fit of the HMA data with coefficients a0= 217.6(3)
GPa, a1=−59(8) GPa/ ˚
A3, and a2= 4.8(6) GPa/ ˚
A6. . . . . . . . . . . . . . . . . . . . . . . 96
6.6 Lattice pressure versus volume of DFT model. The red line is fitting using third-order Birch-
Murnaghan equation of state. The fitting constants (see Eq. 6.13) are c0=−272.53 GPa,
c1= 39969.30 GPa/ ˚
A5/3,c2=−239426.00 GPa/ ˚
A7/3, and c3= 520974.00 GPa/ ˚
A3. Inset:
the error in pressure due to fitting; x-axis is the same as the main figure. . . . . . . . . . . . . 97
6.7 Helmholtz free-energy of DFT model including from lattice, quasi-harmonic (qh), and anhar-
monic (anh) contributions. The anharmonic data are obtained from HMA method. Inset: . . . 98
xi
List of Tables
1.1 The 14 Bravais lattices of monatomic systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Excess free energies of FCC HS solid at density ρσ3= 1.0409, based on simulation of N=
500 spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Relative CPU time for the results reported in Table 3.1. . . . . . . . . . . . . . . . . . . . . . 35
4.1 TIP4P potential model parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Real and reciprocal cutoff parameters of Ewald summation. . . . . . . . . . . . . . . . . . . . 49
5.1 Ensemble averages required for first- and second-derivative properties. ................ 66
xii
Abstract
Developing new materials with specific characteristics relies on the knowledge of their crystalline structure
as well as their properties (e.g. chemical, structural, electronic, magnetic, and optical). Free energy (FE) is
the key quantity from which thermodynamic properties are derived and the stability of crystalline polymorphs
can be verified. Molecular simulation (MS) methods (e.g. Monte Carlo (MC) and molecular dynamics (MD))
provide a reliable tool (especially at extreme conditions) to measure solid properties. However, measuring them
accurately and efficiently (smallest uncertainty for a given time) using MS can be a big challenge especially
with ab initio-type models. In addition, comparing with experimental results through extrapolating properties
from finite size to the thermodynamic limit can be a critical obstacle. Here, we develop novel methods to tackle
these problems.
We first estimate the FE of crystalline system of simple discontinuous potential, hard-spheres (HS), at its
melting condition. Several approaches are explored to determine the most efficient route. The comparison
study shows a considerable improvement in efficiency over the standard MS methods that are known for solid
phases. In addition, we were able to accurately extrapolate to the thermodynamic limit using relatively small
system sizes. Although the method is applied to HS model, it is readily extended to more complex hard-body
potentials, such as hard tetrahedra.
The harmonic approximation of the potential energy surface is usually an accurate model (especially at
low temperature and large density) to describe many realistic solid phases. In addition, since the analysis is
xiii
done numerically the method is relatively cheap which makes it an appealing technique. Here, we apply lattice
dynamics (LD) techniques to get the FE of clathrate hydrates structures. Rigid-bonds model is assumed to
describe water molecules; this, however, requires additional orientation degree-of-freedom in order to specify
each molecule. However, we were able to efficiently avoid using those degrees of freedom through a mathemat-
ical transformation that only uses the atomic coordinates of water molecules. In addition, the proton-disorder
nature of hydrate water networks adds extra complexity to the problem, especially when extrapolating to the
thermodynamic limit is needed. The finite-size effects of the proton disorder contribution is shown to vary
slowly with system-size. This allow us to get the FE in the thermodynamic limit by extrapolating the one
isomer results to infinity and correct for that by the effect from considering proton-disorder measured at a
small system. These techniques are applied to empty hydrates (of types: SI, SII, and SH) to estimate their
thermodynamic stability.
For conditions where the harmonic model fails, performing MS is needed to estimate rigorously the full
(harmonic plus anharmonic) quantity. Although several MS methods are available for that purpose, they do not
benefit from the harmonic nature of crystals — which represents the main contribution and is cheap to compute.
In other words, those “conventional” methods always “start from scratch” even at states where anharmonic
part is negligible. In this work, we develop very efficient MS methods that leverage information, on-the-fly,
from the harmonic behavior of configurations such that the anharmonic contributions are directly measured.
The approach is named harmonically-mapped averaging (HMA) for the rest of this thesis. Since the major
contribution of thermodynamic properties comes from the harmonic nature of crystal, the fluctuations in the
anharmonic quantities is to be small; hence, uncertainty associated with the HMA method is small. The HMA
method is given in a general formulation such that it can handle properties related to both first- and second-
derivatives of free energy.
The HMA approach is first applied to Lennard-Jones (LJ) model. First- and second-derivatives of FE with
respect to temperature and volume yield the following properties: energy, pressure, isochoric heat capacity,
bulk modulus, and thermal pressure coefficient. A considerable improvement in the efficiency of measuring
those properties is observed even at melting conditions where anharmonicity is non-negligible. First-derivative
properties are computed with 100 to 10,000 times less computational effort, while speedup for the second-
derivative properties exceeds a millionfold for the highest density examined. In addition, the finite-size and
xiv
long-range cutoff effects of the anharmonic contribution is much smaller than those due to harmonic part.
Therefore, we were able to get the thermodynamic limit of thermodynamic properties by extrapolating the
harmonic contribution to infinity and fix that with the anharmonic contribution from MS of small systems.
Moreover, the anharmonic trajectory shows better features than the conventional one; it equilibrates almost
instantaneously and data is less correlated (i.e. good statistics can be obtained with shorter trajectory). As a
byproduct of the HMA, the free energy along an isochore is computed using thermodynamic integration (TI)
technique of energy. Again, the HMA shows substantial improvement (50−1000 speedup) over the well-known
Frenkel-Ladd integration (with Einstein crystal reference) method. Finally, to test the method against a more
sophisticated model, we applied it to an embedded-atom-model (EAM) model of iron system. The results show
a qualitatively similar behavior as that of LJ model.
Finally, the method is applied to tackle one of the long-standing problems of Earth science; namely, the
crystal structure of the Earth’s inner core (IC). The extreme condition of the IC (≈330 GPa and ≈6000 K)
made a technological challenge to to investigate iron in lab. Ab initio methods provides a reliable alternative
to study the system under such extreme environment. However, contradictory between experimental and the-
oretical results are still far from resolved. Usually density-functional theory (DFT) is used to determine the
free energy of different iron polymorphs structures (HCP, BCC, and FCC). However, this is only applied at
the 0K limit. On the other hand, using MD simulation with DFT is computationally rather expensive. Here
we tested the HMA approach to get an estimate the free energy of hcp structure along an isotherm of relevant
temperature (6000 K) to the IC state. The method showed a substantial reduction (compared to conventional
method) in uncertainty associated to energy and pressure, and subsequently to the free energy. Results showed
that FE estimation within the meV is possible; thus using our HMA technique is expected to be of great impact
in measuring the FE differences between bcc, fcc, and hcp candidates of pure iron in the IC.
xv
Publications
Chapter 3: Free Energy of Hard Spheres Crystal: A Comparative Study
- S. G. Moustafa, A. J. Schultz, and D. A. Kofke, “A comparative study of methods to compute the
free energy of an ordered assembly by molecular simulation”, J. Chem. Phys. 139, 084105 (2013).
Chapter 4: Free Energy of Clathrate Hydrates: Harmonic Approximation
- S. G. Moustafa, A. J. Schultz, D. A. Kofke, “Efficient harmonic free energy calculations of nonlinear
rigid molecular crystals: I. Theory”, J. Chem. Phys. -In preparation.
- S. G. Moustafa, A. J. Schultz, and D. A. Kofke, “Effects of Finite Size and Proton Disorder on
Lattice-Dynamics Estimates of the Free Energy of Clathrate Hydrates”, Ind. Eng. Chem. Res.
54(16), pp 4487-4496 (2015).
Chapter 5: Direct Measurement of Anharmonic Properties of Solids
- A. J. Schultz, S. G. Moustafa, and D. A. Kofke, ”Mapped average theory”, (2015) - To be published.
- S. G. Moustafa, A. J. Schultz, and D. A. Kofke, “Very fast averaging of thermal properties of crystals
by molecular simulation”, Phys. Rev. X (2015) - In preparation.
xvi
Chapter 6: 6 Thermodynamic Stability of Iron Polymorphs in the Earth’s Inner-Core
- S. G. Moustafa, A. J. Schultz, and E. Zurek, D. A. Kofke, “Thermodynamic stability of Earth’s
inner-core polymorphs using direct measurement of anharmonic contributions”, In preparation.
CHAPTER 1
Introduction
1.1 Molecular Simulation (MS)
Very few problems in equilibrium statistical mechanics have exact solutions [1, 2]. The most com-
mon examples are the ideal gas model, 2Dferromagnetic Ising model, Einstein crystal model, and
harmonic approximation of solids. However, those approximations usually fail to reproduce experi-
mental data over a wide range of thermodynamic states. For example, solids at low temperatures can
be descried (exact at T→0) by the harmonic approximation; however, it fails at higher temperatures
where anharmonicity is non-negligible. On the other hand, experiments can be hard to perform (and
hence expensive) at very extreme (thermal, magnetic, electrical, radioactive, etc.) conditions and/or
toxic environment. Molecular simulation (MS), or sometimes called computer experiment, provides
an alternative route. Given the microscopic model (masses, molecular conformation, interactions,
etc.), MS can be used to predict macroscopic properties (transport coefficients, structural properties,
chemical behavior, etc.) of materials. MS is essentially an exact method; thus, it can serve a twofold
general objective: (1) MS is used to test if the model is adequate to describe the system when results
1
Figure 1.1: Schematic diagram of multiscale simulation.
are compared against experiment, or (2) it can be used to test new analytical theories that use the same
model.
The first MS ever was done in 1953, at Los Alamos laboratory, by Metropolis et al. [3] to study
a dense system of hard disks using (Metropolis) Monte Carlo (MC) method. In 1957, at Livermore
laboratory, Alder and Wainwright [4] first applied of molecular dynamics (MD) simulation to study
dense hard spheres (HS) system. Applications to Lennard-Jones (LJ) [5] potential and more real-
istic models [6] were then followed. Although MC and MD algorithms have hardly changed, the
exponential increase in the computing power made it possible to explore bigger systems and more
complicated models. Rigorous prediction of real macroscopic properties requires multiscale simula-
tions, starting from the electronic picture. Based on time-length scale, multiscale simulations can be
categorized into four scales: (1) ab initio (or first-principles) quantum mechanics calculations (e.g.
2
density-functional theory (DFT) and quantum Monte Carlo (QMC)), (2) atomistic level (e.g. MC and
MD), (3) mesoscale level (e.g. coarse-grained techniques), (4) continuum methods (e.g. finite element
and finite volume methods). However, in this thesis we are mainly focusing on the atomistic-scale
level using MC and MD methods.
Application of MS tools to solid-state systems is one of its most successful achievements. Proper-
ties of new materials that have not yet been synthesized can be predicted on computer. For example,
due to the increasing computing power available nowadays, rigorous ab initio simulations are used to
identify the most thermodynamic stable polymorph structures which can be used as a guide for ex-
perimentalists. Since we are focusing on crystalline systems, we will review some concepts of crystal
structures.
1.2 Crystal Structure
A defining feature of defect-free crystalline solids is the infinitely repeated (in all directions) set of
lattice points. Such arrangement forms what is called Bravais lattice where the points are generated
by set of discrete translation operations given by position vectors R
R≡n1a1+n2a2+n3a3(1.1)
where aiare called primitive (non-collinear) vectors that span the lattice and niare any integer number
(negative, zero, and positive) [7]. The Bravais lattice, however, describes only the geometry of the
periodic images regardless of what species (atoms, ions, molecules, etc.) used to build the system.
Each lattice site is made up by one or more species which are called the basis. Therefore, the crystal
structure is fully specified through
Crystal Structure = Bravais Lattice + Basis
The volume of space that if assigned to each lattice point just fills the space (without overlapping
or voids) is called primitive unit cell; thus, each primitive cell must contain only one lattice point.
However, the space can be filled up by non-primitive unit cells (known as conventional unit cells)
3
Table 1.1: The 14 Bravais lattices of monatomic systems.
4
which are usually chosen to be bigger than the primitive unit cell and to have a desired symmetry.
The common examples are body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal
close-packed (HCP) conventional unit cells.
In 3D-space, crystals are divided into 7crystal systems according to their point groups viz.: tri-
clinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic; in low-to-high point
symmetry order. Based on the arrangement of basis in the primitive unit cell a total of 230 space
groups (translation group + point group) can be generated to describe any possible crystal structure.
Among these 230 choices, a subset of simple 14 Bravais lattices with single basis per lattice point can
be generated. Such crystal structures are called monatomic Bravais lattices and many crystals belong
to them. Table 1.1 shows the 14 lattices with some of them constructed using primitive (P) unit cells
with one basis while others are constructed using conventional unit cells (I, C, and F) with more than
one basis.
1.3 Free Energy Calculations of Solids
Free energy (or thermodynamic potential) is the key quantity in equilibrium thermodynamics. In
addition to its role in assessing thermodynamic stability, materials properties can be derived from it’s
derivatives. In canonical ensemble at constant number of species N, volume V, and temperature T
the Helmholtz free energy Ais given by
A(N, V, T ) = −kBTln Q(N, V , T)(1.2)
where kBis the Boltzmann constant and Qis the canonical partition function which is the link between
the microscopic and macroscopic levels
Q(N, V, T ) = qN
N!Z(N, V, T ) and Z(N, V, T )≡Zexp(−βU (r1, ..., rN))d1...dN, (1.3)
where qis the molecular (translational and rotational) partition function per molecule, which is known
analytically, and Zis the configurational contribution (which is the main focus of this thesis) with
5
β≡1/(kBT),Uis the system intermolecular potential energy, diis an infinitesimal volume in
configuration space given by dri, where dri≡dxidyidziand the integration is carried out over
V. For the case of quantum treatment, a sum over all quantum states will be considered instead of
integration.
The configuration partition function Z(N, V, T )is not given as a simple ensemble average but
rather is probational to the phase-space volume accessible to the system. Thus, the absolute FE can
not be measured directly in simulation. What is being measured usually in MS (and experimentally) is
free energy differences between different states/systems. If the absolute FE is the quantity of interest
then a reference of known (analytically or numerically) FE is used to measure the FE relative to using
thermodynamic integration (TI) [1] or free energy perturbation (FEP) [8] methods.
One of the early references that was first applied to hard discs [9] and spheres [10] is called single-
occupancy cell. Recently, it was modified and applied to hard-sphere [11] and Lennard-Jones [12]
models to estimate the freezing transition. However, this method suffers from weak phase transition
along the integration path and cannot directly extended to rigid-molecule models. Nowadays, Ein-
stein crystal (EC) [13] (or Einstein molecule [14]) reference, in which each particle is tethered to
its nominal lattice site by a harmonic spring, is widely used as a reference system. Although this
method was originally introduced by Frenkel and Ladd to get the FE of hard-spheres system [13], it
was soon extended to both linear [15] and nonlinear [16]molecular crystals. For rigid molecules, FE
was carried out by either adding extra harmonic springs for the rotational dof [17] or by just attaching
number of harmonic springs to some fixed points on the rigid-molecule[18] (at least two for linear
molecules and three for nonlinear molecules). EC method was applied to several ice phases (using
rigid-molecule models) to investigate its phase diagram [18, 16, 19], get the melting point [20, 21] ,
and to get the FE [22] . However, this method is tedious to implement and post-process. The FE of the
rigid-molecule reference is not given analytically, as the case of atomic system; but, rather, it is given
in terms of complex integrals that need to be evaluated numerically using, for example, using Monte
Carlo integration [22], or approximated to an analytical expression [23], in the limit of large EC force
constant, that needs to be evaluated numerically too. In addition, while the EC translational field is
6
independent of the molecular topology, the functional of rotational field depends on the molecular
point-group symmetry. [22] One common choice of the EC rotational field that simplifies simulation
is to have it with the same point-group symmetry as of the molecule itself [17]; although other non-
symmetric choices can be used too [24], but requires additional MC moves [17] (and cannot be used
with MD simulation). Also, the EC spring constants (both translational and rotational) must be opti-
mized to maximize the overlap between the EC and the solid of interest and to avoid quasi-divergence
in the integration path; which can be a numerical challenge to get accurate results [25].
Another reference that is even closer to the real solid [26, 25] (which makes it more accurate than
EC reference [27] is called harmonic crystal (HC), or, sometimes called Debye crystal [27, 25]; in
which the interaction forces are assumed to vary linearly with displacement [28, 29]. For thermody-
namic states at which the anharmonic contribution to the FE is negligible (e.g. at low temperatures),
the harmonic FE can be a good approximation to full system FE. Another advantage of the HC over
EC (or EM) reference is that it is a parameter-free model; all force constants are obtained automati-
cally from the normal-mode analysis of the real solid [25, 27]. In addition, it has a well-known FE
formula that can be computed, systemically, by diagonalzing a Hessian matrix of second-order deriva-
tives of the system energy at the minimum. In practice, the HC reference can be used with TI (or FEP)
methods through either temperature-based physical path (starting from T∼0, at which the system
behaves harmonically, to the desired temperature) [30, 31, 32, 33] or Hamiltonian-based non-physical
path [27, 26, 25]. The temperature-based path has been used with solids of soft-spheres [30, 32, 33],
Nitrogen [34], and different rigid-molecule models of ice [31]. On the other hand, the Hamiltonian-
based method has been applied to Si system [26], flexible-molecule [25] and rigid-molecule [27] ice
models.
Since measuring the full (harmonic+anharmonic) FE of solids is computationally expensive (es-
pecially when first-principle potentials are used; (e.g. DFT)), approximate approaches are usually
employed. One of the commonly used approaches is to neglect, entirely, the entropic term; thus, the
system FE is approximated by just the enthalpic part. Of course this is only true in the 0Klimit
and fails to describe the temperature-dependence behavior of the system. However, it is still widely
7
used with DFT models to predict the structure stability of different systems[35]. A more accurate ap-
proximation is to include the entropic contribution based on full harmonic analysis: or its molecular
local-harmonic approximation. [36, 37, 38, 39] Although it is exact only at 0K, it can give a better
understanding of the structural and thermodynamic properties of solids; especially if the anharmonic
contribution is relatively small.
In this thesis, we introduce very efficient (small uncertainties for a given CPU time) MS methods
to measure the full FE and its derivatives for different crystalline systems.
1.4 Thesis Outline
The thesis is organized as follows. In Chapter 2, we review the definition of free energy and its statis-
tical mechanics microscopic origin. In addition, we review the commonly used methods to measure
absolute free energy of crystalline solids. Finally, the mathematical connection between thermody-
namic/mechanical properties and free-energy derivatives is given. Chapters 3 and 4 are dedicated to
measuring free energy on different systems. In Chapter 3, we provide a comparative study of differ-
ent routes (including known methods) to measure absolute free-energy of hard-spheres crystal using
molecular simulation. Application to a more realistic system, clathrate hydrates, with water molecules
treated as rigid (using TIP4P model) is given in Chapter 4. Efficient lattice dynamics techniques are
used to estimate the harmonic free-energy of such system. A novel method is introduced which al-
lowed us to compute the free energies in the thermodynamic limit considering the proton-disorder
nature of hydrates.
In Chapters 5, we introduce a very efficient novel method, called harmonically-mapped aver-
aging (HMA), to directly measure the anharmonic contributions of solid-state properties leveraging
information from the harmonic behavior of crystal. The HMA method is applied to the well-known
Lennard-Jones pairwise potential. In Chapter 6, we started by testing the HMA method to an EAM
model of iron. Then, the method is used to study the thermodynamic stability of pure iron polymorphs
in Earth’s inner-core using ab initio molecular dynamics simulation. Finally, in Chapter 7 we draw
8
some general conclusions and give possible extensions of our work to different applications.
9
CHAPTER 2
Formalism and Methods
2.1 Free Energy and Partition Function
The “natural” thermodynamic potential of a system in contact with a reservoir of temperature Tand
pressure Pis given by its Gibbs free energy G(T , P )[40]
G(T , P )≡U−T S +P V (2.1)
where Uis the internal energy, Sis the entropy, and Vis the system volume. Usually, it is easier to
deal with a fixed volume, thus it is useful to write Gin terms of the “natural” thermodynamic potential
of a system of fixed Vand T. This is known to be the Helmholtz free-energy A(T, V, N )of a system
on Nmolecules, which is defined as A≡U−T S; hence, Gcan be now written as
G(T , P ) = A(T, V, N ) + P V. (2.2)
While Pcan be directly measured (experimentally and in simulation), this is not the case of the free
energy A; to see why, let’s express Ain terms of its statistical mechanics definition [40]
A(T, V, N) = −kBTln Q(T , V, N),(2.3)
where Qis the canonical partition function of the system which is defined (classically) as an integra-
tion over all the microscopic states
Q(T, V, N) = qN
N!Zexp(−βU (r1, ..., rN))d1...dN, (2.4)
where qis the molecular (translational and rotational) partition function per molecule, which is known
analytically, and the rest is the configurational contribution (which is the main focus of this thesis)
with β≡1/(kBT),Uis the system intermolecular potential energy, diis an infinitesimal volume
in configuration space given by dri, where dri≡dxidyidziand the integration is carried out over
V. For the case of quantum treatment, a sum over all quantum states will be considered instead of
integration.
It can be noticed from Eq. 2.4 that Qis not a given as an ensemble average of a function over
microscopic states; but rather it is directly related to the volume of the phase space available to the
system. Thermodynamic quantities that have the same nature (like free energy and entropy) are called
thermal quantities and can not be measured directly in simulation nor experimentally [1]. Thus, what
is, usually, being measured (and more useful) is the free energy difference between states Aand B.
Moreover, if the absolute free-energy at state Bis the quantity of interest, then we can always choose
A(called “reference” system), to be a system of known free energy (e.g. harmonic crystal) and then
add to it the free energy difference from simulation.
Since the efficiency (shortest time to get uncertainty of certain value) of measuring thermodynamic
properties is the goal of this work, we will give an overview of the different choices of the reference
system, thermodynamic path, and the method of measuring the free energy difference relevant to
crystalline solids. It is important to emphasize here that although the free energy uncertainty depends
on those choices, the absolute free energy should not as it is a state function.
11
2.2 Free Energy Difference Methods
Generally, the FE difference techniques can be classified into (1) density-of-states methods, in which
the free-energy difference is related to the ensemble weight between different systems; and (2) work-
based methods, in which the the free energy difference is measured in terms of average work involved
in transforming the system from the reference to the target [41]. In the present study, we consider
only work-based methods, and two in particular: the “infinitely slow” work process represented by
thermodynamic integration (TI) [1], and the “infinitely fast” process represented by free energy per-
turbation (FEP) [8]. For TI, once the integration path is set, there is not much choice for how it is
implemented—the main degrees of freedom are the step size and the quadrature method. For FEP,
there are important choices to make in how it is implemented, and these issues have been described
in detail elsewhere [41, 42, 43]. In this work we will use an overlap-sampling [44, 45] implemen-
tation of FEP (OS), as originally optimized by Bennett [46]. We will also examine the multi-state
generalization of this due to Shirts and Chodera, and known as MBAR [47].
2.2.1 Thermodynamic Integration (TI)
In TI, ∆A0→1is given by
∆A0→1=Z1
λ=0 ∂U (λ)
∂λ λ
dλ (2.5)
where λis a scalar parameter that switches the system energy from the reference state, U(0), to the
target state, U(1), through a specific path (see Sec. 2.4), and the integrand represents the ensemble
average of the first derivative of the potential energy at a given λ.
It is interesting to consider the relation between the accuracy, precision, and computational effort
involved in a TI calculation. The textbook application of numerical integration assumes that the
integrand function is evaluated with a fixed computational cost, and is free of noise. Thus the focus
of the analysis of these methods is on the accuracy of the result, and how much it is improved by
decreasing the integration step, increasing the number of function evaluations, and thus increasing
the total computational cost. The underlying assumption is that the computational cost necessarily
12
increases in proportion to the number of function evaluations. However, for TI performed in the
context of molecular simulation, this is not the case. The cost to evaluate the integrand by molecular
simulation can be adjusted by trading off precision: consider the evaluation of a general integral F=
Rb
af(x)dx via a Newton-Cotes formula of the form b−a
nPn
i=1 aif(xi)(which in this form does not
exclude unequal spacings of the xi). If each function evaluation is given independently with variance
σ2
i, then the variance of the estimate of the integral will be σ2
F=b−a
n2Pn
i=1 a2
iσ2
i. The integrand
variance σ2
iwill vary with the number miof independent samples used to measure it in a molecular
simulation, according to σ2
i= ˆσ2
i/mi, where ˆσiis independent of mi. The total computational cost
is M=Pmi. For a given nand M, the optimal (minimum σ2
F) allocation of samples is mi=
Maiˆσi/Pkakˆσk. Putting this together, we have for the variance of the integral
σ2
F=b−a
n2
X
i
a2
i
ˆσ2
i
mi
=b−a
n2
X
i
a2
iˆσ2
iPkakˆσk
Maiˆσi
=b−a
n2(Piaiˆσi)2
M
≈Σ2
M(2.6)
where we define Σ≡Rb
aˆσ(x)dx, which is independent of Mand n. The key result here is that the
precision of the estimate of the integral depends only on the total amount of sampling Mapplied
across all the quadrature points, and not on the number of quadrature points per se. This result holds
also if one chooses not to optimize the mi, and uses just a constant amount of sampling m=M/n for
each quadrature point. A significant consideration omitted here is the sampling required to equilibrate
each new state condition; this is difficult to characterize in general, but it will be a lower-order effect,
assuming that the amount of sampling, mi, applied to each state point is not exceedingly small.
The conclusion to take from this analysis is that for TI by molecular simulation, the number of
quadrature points should be increased as needed to ensure that the accuracy of the quadrature is as
good as its precision, i.e., that errors due to the finite integration step do not exceed the confidence
13
limits of the integral. Adding quadrature points while keeping the total computational effort fixed can
be done without significantly affecting the precision of the calculated integral.
2.2.2 Free-Energy Perturbation (FEP)
In FEP, while simulating a system at state 0, perturbations are periodically performed into the state
labeled 1, and averages are taken of the form
∆(βA) = −ln e−∆(βU )0,(2.7)
where ∆(βU )is the difference in U/kBTaccompanying the perturbation. The FE is given by the
logarithm of this average, and is highly prone to bias. To obtain a good result, the simulated system
must sample not only the configurations important to system 0, but also those important to system
1. This can happen only if the important system-1configurations are a subset of those important to
system 0[43, 41, 48]. Often they are not, and the method fails. As a remedy, staging is introduced, to
gradually transition from 0to 1. In this context the path of the transition becomes an issue, as it does
in TI. Further, unless 1-important configurations are already a subset of the 0-important configuration,
the subset requirement can be met only via another staging technique, known as overlap sampling.
Here, an intermediate is defined between stages iand i+ 1, such that its configurations are in the
region of overlap (i.e., the intersection) of those of iand i+ 1. Then the OS is conducted, in separate
simulations, from each stage (iand i+ 1) into the overlap region. This approach is quite robust, not
that difficult to implement, and gives good results. It was originally introduced and optimized by
Bennett [46]. The working equations are
∆Ai→i+1 =−kBTln hγOS /ei+1ii+1
hγOS /eiii
(2.8)
where eiis the Boltzmann factor of state i,exp(−βiU(λi)), and γOS is the overlap weight function,
given by
γOS ≡eiei+1
ei+1 +αei
(2.9)
Here, αis an adjustable parameter, which for optimum performance, according to Bennett, is given
by αopt = exp(−β∆A). Since ∆Ais not known a priori, it has to be estimated self-consistently.
14
This can be accomplished by recording averages for multiple values of αin a single run, or by writing
all the observed perturbation energies to file and solving for self-consistency in the post-simulation
analysis.
If multiple states are indeed used to traverse the path from the reference to the target, there may
be some advantage to performing overlap sampling between all pairs of simulated states, rather than
between just states iand i+ 1. This will be helpful to the extent that the different intermediate states
(say, iand i+ 2) have overlapping configurations, and moreover to the extent that the overlap region
of one pair differs from that of other pairs (thereby providing new information about their relation).
The scheme to do this is known as multistate Bennett acceptance ratio (MBAR), and was presented
and optimized by Shirts and Chodera [47]. With it, the relative FE of state iis given by
∆(βA)i=−ln
K
X
j=1
Nj
X
n=1
ei(xjn)
PK
k=1 Nkek(xjn)/αk
(2.10)
where Kis the number of states, Njis the number of uncorrelated equilibrium samples of state j,
ei(xjn)is the snapshot nof the Boltzmann factor as evaluated at state ifor configurations sampled
at state j, and α’s are constants to be determined. Shirts and Chodera showed that the optimum
estimation of α’s, after applying Bennett’s criteria to all states, is when ln αk=−∆(βA)k. Therefore,
the MBAR method has to be solved using self-consistent iteration methods, and this is best done via
post-simulation analysis on appropriate data that is written to file during the simulation.
If multiple states are indeed used to traverse the path from the reference to the target, there may
be some advantage to performing overlap sampling between all pairs of simulated states, rather than
just between states iand i+ 1. This will be helpful to the extent that the different intermediate states
(say, iand i+ 2) have overlapping configurations, and moreover to the extent that the overlap region
of one pair differs from that of other pairs (thereby providing new information about their relation).
The scheme to do this is known as multistate Bennett acceptance ratio (MBAR), and was presented
and optimized by Shirts and Chodera [47]. With it, the relative FE of state iis given by
∆(βA)i=−ln
K
X
j=1
Nj
X
n=1
ei(xjn)
PK
k=1 Nkek(xjn)/αk
(2.11)
15
where Kis the number of states, Njis the number of uncorrelated equilibrium samples of state j,
ei(xjn)is the snapshot nof the Boltzmann factor as evaluated at state ifor configurations sampled
at state j, and α’s are constants to be determined. Shirts and Chodera showed that the optimum
estimation of α’s, after applying Bennett’s criteria to all states, is when ln αk=−∆(βA)k. Therefore,
the MBAR method has to be solved using self-consistent iteration methods, and this is best done via
post-simulation analysis on appropriate data that is written to file during the simulation.
2.3 Reference Systems
There are several reference systems of known FE (either analytically or numerically) to choose from
to computing free energies of crystals [42]. On the one hand, any system with the same crystal
symmetry as the target (or not [49]) can be used as a reference if its free energy is known from
the literature, or as the result of a different set of calculations. On the other hand, ultimately such
systems have to be related to an analytically tractable reference. In this respect, the choices fall into
two categories. First are references based on non-interacting particles that are forced to adopt the
crystalline symmetry of the target system, either by cell-occupancy constraints [50], or by tethering
them to lattice sites [13]. Second are harmonically interacting models, which can be solved using
the methods of lattice dynamics [28]. This reference has the advantage that it by itself can provide a
quantitative description of the target system at low temperature (except for discontinuous potentials,
e.g., hard spheres). Hoover and co-workers [51] used this reference in the study of soft spheres, and
it has been employed occasionally by others [52, 53, 54, 55].
2.3.1 Einstein Crystal (EC)
In this model, each molecule is interacting, independently, with its respective equilibrium lattice site
through a harmonic potential, all characterized by the same spring constant, kE. Consequently there
is no collective behavior exhibited by this model. The potential energy of an EC of Nmolecules is
16
given by
UE=kE
N
X
i=1
(ri−ri0)2(2.12)
where riis the instantaneous location of molecule iand ri0is its respective lattice position. The
corresponding Helmholtz FE of this system is given, analytically, by [56]
βAE
N=3(N−1)
2Nln βkE
π−3
2Nln N+1
Nln ρ(2.13)
where the last two terms represent the center-of-mass (COM) motion contribution.
2.3.2 Interacting Harmonic Crystal (IH)
In the IH model each molecule is interacting, harmonically, with all other molecules through identical
spring constants kH. In our case, we include interactions with only the twelve first-nearest neighbors.
The harmonic potential energy of this system is given by
UH=1
2kH
N
X
i=1 X
j∈{nbrs}i
(r2
ij −r2
ij,0)(2.14)
where rij,0is the separation vector for iand jfor the minimum-energy (perfect-lattice) configuration.
The corresponding harmonic FE is given by [52]
βAH
N=1
2N
3(N−1)
X
i
ln βλi
2π−3
2Nln N+1
Nln ρ(2.15)
where λiare the eigenvalues of the Hessian and ρis the system density. The summation runs over the
3(N−1) nonzero normal modes (the remaining three modes are zero and correspond to the COM
motion) while the last two terms represent the correction for the CM motion. Unlike the EC system,
collective behavior of molecules is expected due to the intermolecular interactions.
2.3.3 System-based harmonic approximation: Lattice Dynam-
ics (LD)
The IH model assumes identical force constants between all molecules; however, the interaction
strength depends on the intermolecular distances. Such dependence is captured in the standard lattice
17
dynamics (or normal-mode) analysis of crystals [28]. In this method, the force constants between
molecules are driven from the actual potential-energy surface of the system; thus, LD gives a better
representation of the system more than the IH model. The harmonic energy is given by
ULD =1
2X
ij,αβ
Dαβ (i, j)uα(i)uβ(j)(2.16)
where Dαβ (i;j)is the force constant between molecules iand jdue to displacing them by uα(i)
and uβ(j), from their respective nominal lattice sites, respectively. The force constant is defined as
a second derivative of the actual potential evaluated with all atoms at their lattice sites (indicated by
subscript 0)
Dαβ (i;j)≡∂2U
∂uα(i)uβ(j)0
(2.17)
The harmonic free energy is given by the same formula as the IH model (i.e. Eq. 2.15).
For systems with local minimum, the LD model is accurate at low temperature and/or high density,
where fluctuations around lattice points are small (relative to nearest neighbor distances). Thus, it can
be used as a reference system (as we do here) in that region, then using TI (or FEP) the free energy
at other states can be obtained. Although LD is an approximate model, it is usually used, by itself,
to estimate the free energy of crystals. This is usually done with large supercells, systems with
computationally-expensive ab initio potentials (e.g. DFT and QMC), or if thermodynamic path to
harmonic region does not exist. However, at interesting states (e.g. coexistence lines) the harmonic
representation “fails” to accurately estimate free energy. In those cases, temperature-based “effective”
harmonic approximations are used (see next section, 2.3.4).
2.3.4 Self-consistent phonon approximation (SCPA)
The potential energy surface in the harmonic approximation is assumed to have a basin with fixed
curvature and a minimum (at lattice sites). However, in real systems the curvature varies as config-
urations deviate from lattice sites and in some cases the system does not have local minimum (i.e.
unstable at T= 0 with imaginary frequencies). For those cases, the potential energy surface can
18
be assumed to be “effectively” harmonic with temperature-dependent curvature. Such idea is usually
called “temperature-renormalization of the normal-mode frequencies”. Once the effective frequencies
are computed the free energy formula (Eq. 2.15) of harmonic model can be used. Several approxi-
mations are used to treat such temperature effects; for example average force constants method [57],
velocity autocorrelation method [58], variational techniques using Gibbs Bogoliubov inequality [59],
and self-consistent phonon approximation (SCPA) methods. One of the commonly used applications
of the SCPA method with first-principles models is called SCAILD (self-consistent ab initio lattice
dynamics) [60].
2.3.5 Soft-Sphere Crystal (SS)
The classical soft-sphere (SS) model is represented by an intermolecular inverse-power [61] potential.
In our case, the inverse-power parameter, n, is taken to be 12
USS (r) = ǫσ
r12
(2.18)
where ǫand σare the SS model energy and size parameters. A comparative disadvantage of the SS
system is that its FE is not analytically tractable, and can be determined only through application of
molecular simulation. However, the non-trivial properties of the SS model depend on a single dimen-
sionless state parameter, ρσ3(βǫ)1/4, rather than separately on the two groups formed from the density
and temperature, respectively. Thus is it not difficult to establish its properties for all conditions of
interest. This has been done, and the results have been fitted to give the free SS FE in a convenient
mathematical representation [53]. Nevertheless, we perform these calculations again in the present
work, for two reasons. First, we want to include the time needed for these calculations when com-
paring the computational effort required for the various FE methods considered here. Second, in the
present work we use a SS reference in which only the first- and second-nearest neighbor interac-
tions are included; the results given in Ref. [53] are extrapolated to an infinite system, and have no
interaction cutoff. We employed harmonically targeted temperature perturbation (HTTP) with a low-
temperature IH reference to compute the SS FE as a function of ρσ3(βǫ)1/4. Details are as described
19
in Ref. [53].
2.4 Thermodynamic Paths
The selection of the path connecting the reference to the target system is specific to their detailed
nature. However, the paths can be broadly classified into physical (e.g. along Tor V) and non-
physical (e.g. transform the intermolecular potential to a different “non-physical” system) paths.
2.4.1 Physical Path
If the free energy of the reference system is known at some thermodynamic state (e.g. at low Tor
small Vthe system behaves harmonically) then the free energy difference from that point can be
computed, via thermodynamic integration along physical thermodynamic paths, to get the absolute
free energy. For example,the free energy difference, along an isotherm of temperature T, due to
volume change from V0to V1is given by
∆A=−ZV1
V0
PdV. (2.19)
Similarly, the free energy difference, along an isochore of volume V, due to temperature change from
T0to T1is given by
∆(βA) = −ZT1
T0
U
kBT2dT, (2.20)
If all information along the integration path is of interest (for example coexistence line), then the
physical path is an efficient way to get the free energy (in comparison to the non-physical paths, see
next Section).
2.4.2 Non-physical Path
One of the most commonly used path is due to Frenkel and Ladd work [13] (usually called Frenkel-
Ladd (FL) path). It is two-stages path; in the first one, the actual system with potential Uis trans-
20
formed (linearly) into a composite model of the actual plus reference systems (U+Uref ) through a
coupling parameter λ,
U(λ) = U+λUref ,(2.21)
where λgoes from 0to some maximum value, λmax . The free-energy difference between the hybrid
and reference systems can then be obtained using one-stage direct free-energy perturbation. The
value of λmax can be chosen arbitrarily large as long as the final free energy is accurate. However,
an optimum value can always be chosen; for example based on “fair” distribution of computational
efforts among the two stages.
Another path that is suitable for discontinuous potentials (e.g. hard spheres) is called a penetrable
ramp path. This path was introduced by Almarza [62] to evaluate the HS free energy. In this path,
unlike the FL path, the HS behavior is developed slowly through a ramp function as follows
U(λ) = (1 −λ)Uref +λ
1−λUramp (2.22)
where λis the path parameter (such that U(0) = Uref and U(1) = UH S ) and Uramp is define as
Uramp =XX
ij
uij,
uij =
W0(σ−rij), rij < σ
0,otherwise
(2.23)
where W0is the ramp potential strength, which can be adjusted to have a smooth transition path
with the least simulation uncertainty. Again, unlike FL, the temperature (in addition to λand W0)
by itself plays a role in the PR path. The temperature of the reference system can be selected such
that the mean-square-displacement (MSD) of the spheres from their lattice sites is equal between the
reference and the target systems.
21
2.5 Material Properties from Free-Energy
Derivatives
In addition to the pivotal role the free energy plays in assessing the thermodynamic stability of crystal
polymorphs, its derivatives are directly related to the materials properties. This can be seen by writing
the fundamental thermodynamic equation of the infinitesimal change in the Helmholtz free-energy
dA=Udβ+βV X
ij
σij dǫij ,(2.24)
where A ≡ βA and ǫis the strain tensor due to applying a stress tensor σ. Thus, the ensemble average
internal energy and stress are given as first derivatives of the free energy
U=∂A
∂β V
,(2.25)
and
σij =1
V∂A
∂ǫij T
.(2.26)
As a special case, the hydrostatic pressure P(given by−1/3 Trace[σ]) is expressed as a first derivative
with respect to the system volume,
P=−∂A
∂V T
.(2.27)
On the other hand, the second derivatives are related to the isochoric heat capacity CVand elastic
constants Cijkl via
CV≡∂U
∂T V
=−1
kBT2∂2A
∂β2V
,(2.28)
and
Cijkl ≡∂σij
∂ǫkl T
=1
V∂2A
∂ǫij ∂ǫkl T
(2.29)
As a special case of the full elastic constants tensor, the bulk modulus is given by
B≡ −V∂P
∂V T
=V∂2A
∂V 2T
.(2.30)
22
Moreover, the isochoric thermal pressure coefficient γVis given as a derivative with respect to both
Tand Vvia
γV≡∂P
∂T V
=1
T∂2A
∂β∂V −∂A
∂V T(2.31)
Finally, the thermal expansion coefficient αcan be derived from Band γVvia
α≡1
V∂V
∂T P
=γV
B(2.32)
In molecular simulation, those properties are often measured assuming that the coordinates are
fixed (varying uniformly) during temperature (volume) derivatives of FE. This yields the “conven-
tional” formulas (see Table 5.1) usually used in literature. However, if we think of the FE derivatives
as a finite-difference of ∆Awith respect to ∆Tand ∆Vchanges, then the precision of estimating ∆A
depends on the amount of overlap between the coordinates phase-space of the initial and final states
(see Sec. 2.2.2). Thus, mapping (or no mapping) molecular coordinates, according to the conven-
tional method, might result in configurations that are not appropriate the new V+ ∆V(or T+ ∆T)
states. Next section (considered the core of this thesis) is devoted to introducing an alternative coor-
dinates mapping that benefits from the harmonic nature of crystals and, hence, improves measuring
properties, substantially.
23
CHAPTER 3
Free Energy of Hard Spheres Crystal: A Comparative Study
3.1 Introduction and Background
In this chapter, we consider methods for hard (or, more generally, discontinuous) potentials. The
prototype application is the system of hard spheres (HS). Despite the simplicity of the HS model,
its dense phase can approximate the structure of more complex potentials because the short-range
repulsion determines the crystal structure [63]. Further, the HS model can approximate the experi-
mental behavior of the colloids [64, 65]. Our particular interest in methods for this system is more
in consideration of other hard-body potentials, which are of significant interest now as a means for
understanding and predicting the assembly of nanoparticles of various shapes [66, 67, 68, 69].
The intermolecular potential between two HS of the same diameter σis defined as
u(r) =
∞, r < σ
0, r ≥σ
(3.1)
where ris the distance between the sphere centers. Since the HS potential can be only zero or infinity,
the Boltzmann factor of a configuration of NHS, exp(−βUN), can be either one or zero, respectively;
consequently, the configurational partition function is independent of temperature. Therefore, the full
thermodynamic behavior of the HS model can be represented by just a single isotherm. We focus on
the FCC solid at the melting density (ρσ3= 1.0409) as it is the most stable structure.[70, 13, 56]
As mentioned in Sec. 1.3, when molecular simulation is applied to compute free energies, in
practice what is computed is the difference in free energy of the system of interest (the target), with
respect to a reference system. If the free energy of the reference is known, then the difference provides
the absolute free energy of the target (though this is not always necessary, in that a given application
may require knowledge of only the difference, e.g., to determine the relative stability of the target and
reference). Within this general approach, there is considerable flexibility in formulating a free-energy
method. The variables include: (1) the simulation method used to compute the free-energy change
along the path (TI and FEP; see Sec. 2.2); (2) the choice of the reference system (EC, IH, and SS;
see Sec. 2.3); (3) the thermodynamic path followed to join the reference to the target (FL and PR; see
Sec. 2.4).
Here, we emphasize a comparison of the performance of the methods, aiming to determine the
most efficient scheme to evaluate the absolute free energy precisely in the shortest time. Although
some references and paths examined here were used before for the HS model [13, 56, 62, 71], no
computational efficiency and simulation details are available in the literature; accordingly we repeat
all these calculations for the sake of consistency in the comparison. We expect that methods shown to
work best for the HS model will also be good choices for computing free energies of other hard-body
models. We also examine the multi-state generalization of this due to Shirts and Chodera, and known
as MBAR [47].
3.2 Simulation Details
Standard canonical-ensemble (N V T ) Monte Carlo (MC) simulations[1] were used to compute the
averages specified in Sec. 2.2. All simulations were performed for the FCC crystal at the density
of the target system (ρσ3= 1.0409). For the HS and SS systems, we include interactions between
25
only those particles that are first- or second-nearest neighbors on the perfect lattice (second neigh-
bors are separated by rcut = (4/ρ)1/3σ= 1.5663σfor the density studied here); only first-neighbor
interactions are considered in the IH reference.
For each simulation (i.e., each point on the thermodynamic path), a total of 108proposed MC
trial steps were conducted, beyond an initial 50Nequilibaration trials performed before collecting
any data. Contributions to the ensemble averages were taken after every Ntrials. The samples were
sub-averaged into 100 blocks that were used to provide error estimates, with the method of Kolafa[72]
applied to account for any correlation between adjacent block averages. Thus, for example, for the
N= 500 system for which we report the most detailed results, we collected 108/500 = 200,000
data samples, and for the error analysis computed 100 block sub-averages, each formed from 2000
samples taken across 106MC trials. Reported confidence limits represent one standard deviation of
the mean.
In all calculations the center of mass was fixed to avoid divergence associated with drifting the
whole system; this was achieved by moving two HS at a time in each proposed trial move. A fixed
cubic box of edge L= (N/ρ)1/3σwas used, in which Nparticles were positioned, at the beginning
of simulation, on the equilibrium FCC sites corresponding to the box size. In all calculations, standard
periodic boundary conditions were employed.
The step size of the multistage TI or OS depends on the path. For the PR path, we found that
50 steps of size dλ= 0.02 (where λvaries from 0 to 1) for all references is reasonable (we tried
dλ = 0.01, but we got similar results). For the FL path, the number of path steps was selected as the
fewest needed to obtain a result of accuracy that is greater than the precision of the calculated free
energy; the required number is approximately 50, and the values used are given in Sec. 4.4. The FL
steps give a consistent increment in tfrom ln cto ln(c+λm), and the values of cand λmemployed
here are also given in Sec. 4.4. The reference temperature of the FL path was set to unity, while the
temperature for the PR path was optimized using the MSD criteria described above (which differs
based on the reference system, as shown in the results Section 3.3.1).
We screened the choices by first performing MC simulations for a system of 500 HS to identify
26
the most efficient reference and path combination; then calculations were performed on the selected
combination for different system sizes (N= 256,864,1372,2048,2916,and 4000) to evaluate the
HS FE in the thermodynamic limit (N→ ∞) through extrapolation.
The FE of SS reference was computed in a separate set of MC simulations, using the HTTP
method.[53] Starting from kBT /ǫ = 0.01 to kBT/ǫ = 0.55 with step of 0.01, a polynomial fitting
was done on this temperature range so that at any desired temperature the FE can be evaluated.
3.3 Results and Discussion
3.3.1 Reference Temperature for PR Path
For the PR path, we set the reference system temperature based on equating the MSD of both the
HS and reference systems at the same conditions. The MSD for the EC and IH references is known
analytically and varies linearly with T;[73] on the other hand, there is no analytical form for the SS
MSD; but it should behave harmonically at low temperature. The MSD versus temperature for each
of the three references is presented in Fig. 3.1. The temperature corresponding to the intersection of
the MSD of HS and each reference system is kT /ǫ = 0.0117,0.1150,and 0.4100 for EC, IH, and SS
references, respectively, and these determine the temperatures used for the isothermal paths taken for
the corresponding FE calculations. To check the suitability of the MSD criterion we performed FE
calculations (not shown here) also at reference temperatures below and above these temperatures, and
we found that the simulation error is minimized at the MSD-based temperatures.
3.3.2 Free Energy Results for Finite-Size Systems
Table 3.1 summarizes free energies computed using the assorted combinations of reference system,
thermodynamic path, and FE method for the N= 500 system. The excess FE per particle with respect
to the ideal gas (βAex/N =β(A−AIG )/N; where βAIG/N = ln ρ−1 + ln(2πN)/2N) is modified
27
Table 3.1: Excess free energies of FCC HS solid at density ρσ3= 1.0409, based on simulation of N= 500 spheres.
For both TI and OS, the FE results are given for the PR and FL paths, using EC, IH, and SS references. The digit(s)
in parentheses is the uncertainty of the last digit(s) in the reported value, represented by the 68% confidence limit. The
number of MC trials is 108for each simulation, and the total number of simulations for each free-energy calculation is
given in the row labels “Path steps”. This free-energy uncertainties are each scaled to a common length of 50 path steps
for the purpose of comparison (multiplying by (Path steps/50)1/2), and tabulated with higher precision in the row labeled
“Error”. Temperature (kBT /ǫ)and path parameters (dimensionless) are given as indicated.
TI method
Path PR FL
Reference EC IH SS EC IH SS
βAex /N + ln N/N 5.90851(16) 5.90844(12) 5.90830(6) 5.9082(2) 5.9080(3) 5.9082(2)
Error×1041.6 1.2 0.61 2.6 2.4 2.2
Temperature 0.0117 0.115 0.410 1.0 1.0 1.0
Path steps 50 50 50 55 41 50
Path parameters
W0= 0.8W0= 8.0W0= 60.0 ln c= 2.0 ln c= 1.0 ln c= 1.0
dλ= 0.02 dλ= 0.02 dλ= 0.02 λm= 1628.6λm= 145.7λm= 21.79
OS method
Path PR FL
Reference EC IH SS EC IH SS
βAex /N + ln N/N 5.90852(18) 5.90831(14) 5.90825(5) 5.9086(3) 5.9079(3) 5.9080(3)
Error×1041.8 1.4 0.46 3.2 2.9 2.5
Temperature 0.0117 0.115 0.410 1.0 1.0 1.0
Path steps 50 50 50 55 41 47
Path parameters
W0= 2.0W0= 20.0W0= 80.0 ln c= 2.0 ln c= 1.0 ln c= 0.2
dλ= 0.02 dλ= 0.02 dλ= 0.02 λm= 1800.7λm= 178.5λm= 19.70
28
0.0110
0.0112
0.0114
0.0116
0.0118
0.0120
0.110
0.112
0.114
0.116
0.118
0.120
kT/ε
16.75 17 17.25 17.5 17.75 18
MSD*1000
0.400
0.405
0.410
0.415
0.420
EC
IH
SS
HS MSD (17.433)
Figure 3.1: M SD vs. Tfor different reference systems (EC, IH, and SS, as indicated) at ρσ3= 1.0409 and
N= 500. The dotted vertical line represents the MSD of the HS system at the same conditions.
by adding ln N/N to remove the logarithmic leading dependence on N, so that βAex/N + ln N/N
scales as N−1.[56]
To show that the FE computed by the twelve combinations of methods are mutually consistent,
we compute the statistic,
c≡v
u
u
t 1
M
M
X
i=1
(Ai−Aavg )2
σ2
i+σ2
avg !(3.2)
where Mis the number of data sets (12 in our case), Aiand σiare the FE and uncertainty, respectively,
resulting from by a specific combination of choices, and Aavg and σ2
avg are
Aavg =PM
i=1 (Ai/σ2
i)
PM
i=1 (1/σ2
i)(3.3)
and
σ2
avg ="M
X
i=1 1/σ2
i#−1
(3.4)
For the results to be mutually consistent, cshould be of order of 1.0. Using the data given in Table
3.1, c= 0.94, indicating that the variation in the FE estimates from the different methods is consistent
with the stochastic error estimates.
29
We notice that the uncertainty (as scaled to 50 simulations; see the Error row) obtained from both
TI and OS are very similar when compared for the same reference and path. More important, we find
that the combination of the SS reference and the PR path exhibits the least uncertainty (at least 2 times
smaller) of all the combinations. This should not be surprising, as the SS behavior is expected to be
closer to the HS due to its harder repulsion (compared to IH) and its collective motion (compared to
EC). On the other hand, the uncertainty associated with the FL path is independent of the reference
system, suggesting that the total uncertainty in this case is dominated by the phasing in of the HS
potential along the path, more than the choice of the reference.
To check if the multistate MBAR method has advantage over the pairwise OS method we con-
ducted a separate set of simulations (apart from the results given in Table 3.1) for both methods. It
is most convenient to base the analysis on uncorrelated data, rather than attempting block averaging
and compensating any residual correlation explicitly (as we do for Table 3.1 results), so we generated
data for the analysis using a sampling interval of 105steps; we find that the correlation between adja-
cent samples is statistically zero. Both methods used the same data set as an input, enabling a direct
comparison. The raw data from the MC simulations were processed for MBAR using a Python script
developed by Shirts and Chodera.[47]
For this comparison, we consider a cubic box of 500 HS, using SS as the reference and the PR path.
Using 109MC steps (i.e. 104uncorrelated samples), we obtained excess free energies of 5.90839(12)
from OS and 5.90841(12) from MBAR. This result indicates that adding extra perturbations (i.e.
MBAR) does not improve the uncertainty of HS FE calculations, in this application. The value mea-
sured in the perturbation from the ito i+ 1 state is strongly correlated with that from perturbations
from iinto subsequent states (i.e.,i+ 2,i+ 3,etc.), and this significantly diminishes the information
contributed by these extra perturbations. We observe, in particular, that the correlation between the
i→i+ 1 and i→i+ 2 perturbations to be of the order of 0.9 for all states on the perturbation path.
The performance observed here is not inconsistent with other studies,[74] which have found that the
relative performance of MBAR and OS depends on the nature of the free-energy calculation, as well
as implementation details.
30
0 0.2 0.4 0.6 0.8 1
λ
-0.4
-0.3
-0.2
-0.1
〈∂Uramp(λ) / ∂λ〉λ
00.5 1
λ
-0.001
0
0.001
0.002
yN - y4000
N=864,2916
N=500
Figure 3.2: Integrand of Eq. 2.5, ∂A/∂λ, for the SS reference and the PR path at density ρσ3= 1.0409.
Results are plotted for N= 4000, but results taken for all other Nwould not be discernible on the larger figure.
Inset shows for three system sizes (N= 256 (black), 864 (red), and 2916 (blue)) the difference at each λwith
the corresponding value for the largest system (N= 4000).
Given the observed equivalence of OS and TI, we choose TI for all of the subsequent analysis,
due to its simplicity to implement and post-process. Further, we will focus on using the SS reference
and PR path combination, as they show the lowest uncertainty in estimating the solid HS FE.
3.3.3 Finite-Size Effects: Thermodynamic Limit Results
The integrand for the PR+SS+TI calculation (see Eq. 2.5) is shown for different system sizes in Fig.
3.2. The results are very precise, with error bars that are not visible on the scale of the figure. The
curve is not straight, but it does vary smoothly at all values of λ. The integrand is almost completely
independent of N—results for the seven system sizes would not be distinguishable on the scale of the
full graph. We use the inset figure to uncover the small finite-size effects. This shows the integrand
relative to the value measured for the largest system. For clarity, only three values of Nare plotted,
and it is seen that for only the smallest size does the difference from the largest simulated system
exceed the error bars. The small finite-size effects can be made visible also by examining the integral
31
0 0.001 0.002 0.003 0.004
1/N
-0.6688
-0.6684
-0.6680
-0.6676
-0.6672
-0.6668
β(∆A/N)SS→HS
[β(∆A/N)SS→HS]fit = - 0.66856(3) + 0.398(18)/N
Figure 3.3: Finite-size effect of the FE difference between SS and HS systems at the melting condition (ρσ3=
1.0409).
itself, as shown in Fig. 3.3. Here, we present the FE per particle, relative to the SS reference, at
different system sizes (plotted as 1/N). The slope is 0.40(2), which is more than ten times smaller
than the slope of the full FE as reported by Polson et al.,[56] who used EC as a reference, or our
own results using EC or size-dependent IH and SS references (Fig. 3.4). This result suggests that
the finite-size effect is dominated by the SS reference contribution, which is absent in Fig. 3.3. This
point is made in Fig. 3.4, where we examine the finite-size effects in terms of the full excess FE,
highlighting the role of the finite-size effects brought by the reference. We present results obtained
using TI+PR with SS, IH, and EC references, as well as results reported previously by Polson et al.
[56] using the TI+FL+EC combination. We consider results obtained both by size-dependent as well
as infinite-system IH or SS references.
The nature of the finite-size effects displayed in Fig. 3.4 for these HS calculations is consistent
with finite-size effects observed for SS (for different values of the inverse-power exponent),[52, 53]
and for a model of molecular nitrogen (N2).[75] If the reference system has some semblance to the
target, and in particular if it is formed from interacting particles and thus exhibits cooperative behavior
(e.g. IH and SS systems in the present case), then the finite-size effects it exhibits for the FE will
roughly mirror the finite-size effects in the target. Consequently, the FE difference—the difficult
32
0 0.001 0.002 0.003 0.004 0.005
1/N
5.890
5.895
5.900
5.905
5.910
5.915
5.920
5.925
βAex/N + ln N/N
Polson et al.
SS ref.
IH ref.
EC ref.
SS ref.: 5.91901(3) - 5.34(2)/N
IH ref.: 5.91906(6) - 5.34(3)/N
EC ref.: 5.91921(10) - 5.41(5)/N
IH ref. (HybridInf): 5.91906(6) + 0.62(3)/N
SS ref. (HybridInf): 5.91901(3) + 0.40(2)/N
Polson et al.: 5.91889(4) - 6.0(2)/N
Figure 3.4: Finite-size effect of the excess FE at the melting state (ρσ3= 1.0409). Lines for the EC, IH, and
SS references were computed by adding the FE of the corresponding size-dependent reference to the (size-
dependent) FE differences computed via TI on the PR path. Results by Polson et al.[56] give similar results for
the TI+FL+EC method. Lines labeled HybridInf are computed using a size-dependent TI+PR integral with the
FE for an infinite-system reference.
calculation that is the focus of this work—will be largely system-size independent. Tan et al. [52]
proposed a hybrid approach to exploit this behavior. In this hybrid method, the free energy of the
reference is extrapolated to the infinite system, and this (rather than a size-dependent reference value)
is added to the finite-system FE difference to obtain the target-system FE. The absolute FE computed
this way will have a system-size dependence with slope given by that of the FE difference, which
is nearly flat (Fig. 3.3). The advantage in doing this is that the FE of the reference is more easily
determined as a function of system size, compared to the calculation of the FE difference, which
involves TI or OS and molecular simulation. Thus we can obtain a good estimate of the target-
system FE in the thermodynamic limit while conducting small-system molecular simulations. Fig.
3.4 includes results (labeled HybridInf) that demonstrate the idea for both SS and IH references. As
shown there, although the finite-size effect of the FE difference using IH reference is quite small,
its slope is a little larger than that of the SS reference. We did not apply this technique to the EC
33
reference because it has no finite-size variation, so its “hybrid” implementation would not differ from
the curve already given in the figure.
We notice in Fig. 3.4 that while the results from the size-dependent EC, IH, and SS references
are mutually indistinguishable, they are considerably apart from the results of Polson et al.. The
difference is likely to be due to the use of different box shapes for the MC simulations in the re-
spective studies. Still, the values extrapolated to 1/N →0differ. The confidence limit given on
the extrapolated value reported by Polson et al. is much tighter than almost all of the error bars on
their size-dependent data, and the high precision in this extrapolation seems to owe to the high preci-
sion of the result from their largest system simulated. This point notwithstanding, their data are not
inconsistent with an extrapolation that agrees with the values we report.
The extrapolated excess FE at the thermodynamic limit, based on our calculations using SS,
IH and EC references, is 5.91901(3), 5.91906(6), and 5.91921(10), respectively; these results are
close to values reported previously: 5.924(15),[50] 5.9226(10),[13] 5.91889(4),[56] 5.91930(11),[62]
5.9194(5),[76] 5.9194(2),[77] and 5.9188(6)[71] (a small adjustment is applied to the literature val-
ues, where needed, to bring them to the density examined here, ρσ3= 1.0409).
3.3.4 Computation Efficiency
While the SS reference with PR path showed the least uncertainty with respect to number of MC
cycles, it is worthwhile also to compare the efficiency of computations with respect to the CPU time.
For this purpose, we estimate the amount of CPU time trequired by each method to produce a result
to the same precision as the TI+PR+SS method (our base method for comparison). The comparison
is slightly complicated by the need to compute the SS FE for methods that use it as a reference. One
might be interested in scenarios in which the SS FE either is taken as given, or, in the interest of fair
comparison of total computation effort, the time required for its calculation, tref, is included. We will
present both comparisons. Regardless, we always include the contribution of the SS reference FE to
the total stochastic error, according to σ2
tot =pσ2
ref +σ2
∆, where σ∆is the error reported in Table
3.1 (for the actual number of path steps used); for EC and IH, effectively we take σref = 0,tref = 0.
34
Table 3.2: Relative CPU time for the results reported in Table 3.1. Tabulated are the CPU times required to get the same
total error as resulted from the TI+PR+SS base case, divided by the base-case CPU time tbase = 15.7hours. Values in
parentheses for methods using a SS reference show relative timings that exclude the contribution tref needed to obtain the
SS reference FE. Values are reported to two digits of precision, but the second digit is not highly significant.
TI method
Path PR FL
Reference EC IH SS EC IH SS
t/tbase 2.3 1.4 1.0 (0.5) 6.0 5.9 5.2 (5.1)
OS method
Path PR FL
Reference EC IH SS EC IH SS
t/tbase 3.5 2.1 0.9 (0.4) 10. 10. 7.5 (7.4)
We take σref as fixed, and assume that adjustments of CPU time affect only σ∆as needed to make
σtot equal to that for the base method (7.3×10−5). CPU time is estimated assuming that tσ2
∆is an
invariant with t.
Where needed, we compute the SS FE using the HTTP method.[53] For the case of PR path, we
obtained the SS FE (βA/N) at T= 0.41 with uncertainty σref = 2.80×10−5using 7.9CPU hours. For
the FL path, the FE is required at a lower temperature, T= 0.0459 (=1/λmax; where λmax = 21.79, see
Table 3.1), so only 1.6CPU hours were needed to yield a result with uncertainty σref = 1.90 ×10−5.
An additional 7.8 CPU hours were required to perform the TI calculation following the PR path to
obtain the TI+PR+SS base-case FE reported in Table 3.1. All simulations were performed on Intel
Core2 Quad Q9550 Yorkfield 2.83 GHz processor, and all results are reported for the N= 500
system.
Table 3.2 presents the comparison. All times are divided by the CPU time for the TI+PR+SS base
case, which includes tref. Even when including computational effort of the SS reference, the PR path
and SS reference is still the most efficient way to get the HS FE, requiring the least computational
35
time for a given precision. These results reinforce the conclusions taken from Table 3.1. We also
notice that the time required for TI method is marginally lower than that of the OS method at the
same path and reference combination, except for the PR+SS case. Also, the time needed for the PR
path is always shorter than that of the FL path (whatever reference used). The advantage of the PR+SS
methods is even more striking if one allows the SS FE as a given—in such a case this reference/path
combination requires 10-20 times less computation than approaches based on the FL path.
3.4 Conclusions
This work presents a comparison of the efficiency of molecular simulation methods for measuring
absolute free energies of hard-particle models. The comparison is based on the HS FCC crystal at the
melting condition, but it is reasonable to expect the conclusions are general to hard-body models, or at
least to rigid, convex bodies. The study employs various combinations of references (Einstein (EC),
interacting harmonic (IH), and r−12 soft-sphere (SS) crystals), switching paths (Frenkel-Ladd (FL)
and penetrable ramp(PR)), and free-energy methods (thermodynamic integration (TI) and overlap-
sampling (OS) free-energy perturbation).
The study shows that, in the best cases, OS and TI are roughly equivalent in efficiency, with
a slight advantage to TI. When attempting to obtain very accurate free energies by evaluating the
difference along a thermodynamic path, the choice between TI and OS is a matter of convenience and
preference. For high-accuracy applications, both perform about equally well in producing a value to a
given precision, all else being the same. If high accuracy is not a priority, TI can provide a rough result
with as few as two simulations, while OS for the same system could fail catastrophically. However,
it still not a good practice to apply TI without some means to gauge its accuracy, particularly by
examining how the result changes with number of quadrature points. We also examine the multistate
Bennett acceptance ratio (MBAR) method, and find that it offers no advantage for this particular
application.
The PR path shows advantage in general over FL, providing results of the same precision with 2 to
36
9 times less computation, depending on the choice of a common reference. The best combination for
the FL path is TI+EC, which is how the FL method is usually implemented. For the PR path, the SS
system (with either TI or OS) proves to be most effective; it gives equivalent precision to TI+FL+EC
with about 6 times less computation (or 12 times less, if discounting the computational effort required
to establish the SS reference free energy).
The results of this study for a hard potential reinforces the observations regarding system-size de-
pendence made previously for soft-interaction models. The difference in FE between a target system
and a reference formed from interacting particles (IH or SS) is nearly independent of system size.
The hybrid technique for handling finite-size effects is thus the method of choice: the finite-system
FE difference is added to the extrapolated (1/N →0) absolute FE of the reference system. This
method provides a good and rapid estimation of the infinite-size system FE using only one simulation
of a relatively small system size. A second calculation at a different system size can be performed to
confirm the size independence, if needed.
The methodology examined here is readily extended to more complex hard-body potentials, such
as hard tetrahedra. [67] In such an application, the hard potential defining the shape of the hard
body is transformed into a softer repulsive form. The PR path with OS or TI can be used compute
the FE difference between the hard and soft potentials, and HTTP can then be used to lower the
temperature of the soft potential to connect it to an interacting harmonic reference. The ability to
perform such calculations for small systems, relying on the reference to capture the finite-size effects,
can make such calculations increasingly practical, and thereby enable a rigorous evaluation of the
relative stability of different candidate structures.
37
CHAPTER 4
Free Energy of Clathrate Hydrates: Harmonic Approximation
4.1 Introduction and Background
Clathrate hydrates are crystalline compounds formed from water and small hydrophobic solutes (e.g.
methane) in which the water molecules form a lattice of hydrogen-bonded polyhedral cages, each
surrounding one or perhaps a few of the solute (guest) molecules.[78, 79] Clathrates are important
for both technological and environmental reasons. From a technological standpoint, the formation
of clathrates is a major concern in the oil and gas industry, where undesired formation in pipelines
produces solid plugs that halt flow. Consequently, significant effort is directed toward flow assurance
by prevention or control of the formation of clathrate hydrates. A more positive technological issue
is in the exploration of clathrates as a means for storage and transport of gaseous fuels, particularly
methane and even hydrogen,[80, 81, 82] or for sequestration of pollutants and greenhouse gases, espe-
cially carbon dioxide.[83] Natural stores of methane in permafrost or seafloor hydrates are huge,[84]
and recovery of this resource, particularly in conjunction with a complementary process to sequester
CO2, could provide a long-term energy source to meet worldwide demand. Other uses envisioned for
clathrate hydrates include gas separation,[85] desalination,[86] and refrigeration.[87, 88]
Clathrate compounds are nonstoichiometric, in that some of the cages may be vacant or have
multiple occupancy at equilibrium. The lattice of empty cages is mechanically stable, but there is
a minimum occupancy required for thermodynamic stability. Hydrates appear in several structures,
each of which combines two or more cage types to form a unit cell, and differing in the size, shape,
and number of cages in the repeat unit. The most common structures[79, 89] are: structure I (sI,
with 2 small + 6 large cages formed from 46 water molecules in the unit cell);[90] structure II (sII,
16+8 cages from 136 water molecules);[91] and (less common) structure H (sH, 3+2+1 cages from 34
waters).[92] The structure adopted by a clathrate hydrate generally depends on the size of the solute
molecule.
Given their practical importance, it is not surprising that these compounds have received an in-
tense amount of attention from the research community. They have been studied by experiment for
centuries.[78] It has been found that thermodynamic behavior is for many purposes adequately de-
scribed by the van der Waals-Platteeuw (vdWP) model.[93, 94] This is a statistical-mechanical theory
that treats the hydrate as individual cages housing guest molecules that do not interact across cages. It
further assumes single occupancy of the cages, that occupancy of one cage has no effect on others, and
that quantum effects can be neglected. Various extensions have relaxed one or more of these assump-
tions. However, selective removal of approximations might in fact be counterproductive, as molecular
simulation studies[95, 96, 97] have found that the vdWP model gains some accuracy through a can-
cellation of errors. The ability of molecularly detailed modeling to provide insights such as this,
coupled with many advances in methods, models, and computing hardware in recent years, has led
to increasing interest in these approaches as a route to understand, predict, and manipulate clathrate
behavior.
Of central importance to all modeling studies is knowledge of the thermodynamic phase behavior
of the system being modeled. Often this information is useful as an end in itself, for example when
the study aims to gauge the number of guest molecules that can occupy the cages of a given clathrate
structure under different conditions.[98, 99, 100, 101, 102, 103] Knowledge of the phase behavior is
39
important also as a backdrop for other studies, such as those relating to nucleation and growth.[104]
Interpretation of molecular simulation data for these processes cannot be made conclusive without
knowledge of the stability boundary and thermodynamic forces driving the behavior.[89, 105] Inas-
much as thermodynamic properties for different molecular models do not in general coincide with
each other (or with experimental behavior), it is necessary to gather these data for the specific molec-
ular model underlying the (for example) nucleation studies.
Free-energy calculations in molecular simulation are often problematic or computationally expen-
sive, although they are becoming routine. Still, applications to clathrate hydrate structures introduce
special difficulties, and relatively few rigorous absolute free-energy calculations have been conducted
for these systems[106, 107, 95, 108, 109, 110, 111] (we exclude here free-energy calculations and
grand-canonical simulations that yield cage-occupancy statistics, given the clathrate structure). Much
more common are studies employing harmonic[112, 113] and quasi-harmonic[114, 115, 116, 99,
100, 101] treatments based in lattice dynamics. These approaches are approximate, and are lim-
ited to the range of (low) temperatures where the harmonic approximation is correct. Nevertheless,
they represent an important component in the collection of methods available for the modeling of
clathrate systems, and indeed crystalline water (ice)[117, 118, 119, 120] phases in general: they are
much faster than molecular simulation in yielding values of thermodynamic properties; they have
no difficulty capturing the quantum nature of oscillation that prevails at low temperature; and they
provide a suitable starting point for more rigorous molecular simulation studies.[42] In particular,
thermodynamic integration from a low-temperature state, where the harmonic treatment is accurate,
provides a convenient route to the free energy at an arbitrary condition of interest. Moreover, it has
been observed[32, 33, 34, 121] that finite-size effects inherent in molecular simulations of solids are
largely due to the harmonic behavior of the crystal. This means that molecular simulation methods
that isolate anharmonic effects can be applied to measure properties for relatively small systems with-
out incurring finite-size errors, and the infinite-system harmonic behavior can be added afterward to
yield the full behavior in the thermodynamic limit.
The assumption that the energy can modeled using a harmonic expansion about a minimum-energy
40
configuration is not the only approximation inherent in practical lattice-dynamics calculations of hy-
drates and ice phases. Given the foundational role held by lattice dynamics methods for the study
of these systems, it is worthwhile to examine some of these other approximations. First, it is well
known that crystalline structures of water are often proton-disordered, meaning that the orientations
of the water molecules are not uniquely specified as part of the crystal structure.[122] Within cer-
tain constraints, there are many possible structural isomers, each differing in the way that the water
molecules are each oriented. The usual practice is to select a low-energy isomer with near-zero dipole
moment, and perform lattice dynamics calculations using this single structure. Sometimes multiple
structures are used, and properties are averaged in a simple manner. In the present work, we examine
the suitability of such approaches. Second, we consider finite-size effects and how the free energy
extrapolates to the thermodynamic limit. Such understanding is needed if we aim to use the harmonic
system as a reference that captures the large-system behavior when coupled to (small-system) molec-
ular simulations that correct for anharmonic effects. Finally, some studies[36, 37, 39] make use of a
more approximate local-harmonic approach, which is a mean-field treatment in which each molecule
oscillates in a field imposed by the others, effectively forming an Einstein crystal with anisotropic
force constants determined by interactions with the other molecules. The suitability of this approxi-
mation is easily tested as part of the overall study.
4.2 Theory and Methods
4.2.1 Lattice Dynamics of Rigid Molecules
In this section we review the framework for estimating the free energy, under the assumption that
the the intermolecular interactions can be approximated by a harmonic potential. We develop the
methods for application to a single proton-disordered structure, which we take as given. Approaches
for accounting for proton disordering will be described in Sec. 4.2.2.
The normal modes of motion of flexible molecules are classified into “external” (translation and
41
rotation of the whole molecule) and “internal” (internal vibrations and bending). In this work we use a
rigid model for the water molecule, for which only external modes are present. Accordingly, we focus
our review on the harmonic analysis of the external modes of non-linear rigid molecules. The concepts
and methods are all well established,[123, 28, 124, 125] so we will not attempt to be complete. Instead
we will highlight the key steps in the treatment needed to understand the contributions made in this
work, while pointing out some novel features in our implementation.
For non-linear rigid molecules, in addition to the three translational degrees of freedom (dof),
three angular dof per molecule are required to fully describe the system configuration. The linear
and angular displacements of a molecule κin cell lalong αdirection will be denoted by ut
α(lκ)and
ur
α(lκ), respectively. ut
α(lκ)represents the translational deviation of molecule κ’s center of mass
(CM) from its equilibrium position. Concerning rotation, there are several choices for an appropriate
definition of ur
α(lκ); the only requirement is to be able to use them to expand the potential energy.
The most common choice is the angle of rotation around αaxis (adopting the right-hand rule) with
respect to its nominal orientation in the crystal. For small rotations, the rotation can be decomposed
into three rotations around x,y, and zaxes (θx,θyand θz), respectively.
In the sums that follow, κgoes over all nmolecules in unit cell l;αand βgo over all coordinate
directions, x, y, z;igoes over tand rfor translation and rotation; and lgoes over Ncreplicas of
the unit cell (all having the same proton-disordered structure), forming a supercell. The number of
molecules in the supercell, N, equals nNc. Primes on each of these indices are used to form a second
set of sums.
The potential energy Uincludes interactions among all of the Nmolecules of the supercell, and
may (as it does here) include interactions of these Nmolecules with an (in principle) infinite periodic
system surrounding it (evaluated in practice via a combination of lattice and Ewald sums). One of our
interests in this work is how the free energy approaches the thermodynamic limit with increasing N
when Uis estimated via a harmonic potential. Accordingly, Uis approximated via a Taylor expansion
42
in molecular displacements (both linear and angular) as follows: [124]
U≈U0+1
2X
lκα
l′κ′β
∂2U
∂ut
α(lκ)∂ut
β(l′κ′)!0
ut
α(lκ)ut
β(l′κ′) + 1
2X
lκα
l′κ′β
∂2U
∂ur
α(lκ)∂ur
β(l′κ′)!0
ur
α(lκ)ur
β(l′κ′)
+1
2X
lκα
l′κ′β
∂2U
∂ut
α(lκ)∂ur
β(l′κ′)!0
ut
α(lκ)ur
β(l′κ′) + 1
2X
lκα
l′κ′β
∂2U
∂ur
α(lκ)∂ut
β(l′κ′)!0
ur
α(lκ)ut
β(l′κ′),
(4.1)
which is truncated at second order. We omit the terms linear in ui
α(lκ)because the derivatives are
evaluated with all molecules at their respective equilibrium (minimum energy, U0) positions and ori-
entations, and consequently all first derivatives with respect to both linear and angular displacements
(i.e. forces and torques) vanish. We are left here with only the second-order contribution, representing
the harmonic approximation to the energy, Uharm (defined in excess of U0). The harmonic energy can
be written more compactly in terms of generalized force-constant coefficients, φii′
αβ(lκ;l′κ′), defined
by
φii′
αβ(lκ;l′κ′) = ∂2U
∂ui
α(lκ)∂ui′
β(l′κ′)!0
,(4.2)
as follows
Uharm =1
2X
ll′,κκ′
αβ,ii′
φii′
αβ(lκ;l′κ′)ui
α(lκ)ui′
β(l′κ′).(4.3)
With the energy approximated as in Eq. (4.3), we are left with a system of coupled harmonic
oscillators, for which the dynamics and thermodynamics can be evaluated analytically. The conven-
tional approach is to solve for the dynamical behavior, which yields a set of frequencies ωjfor what
are effectively a set of independent harmonic oscillators, each describing some collective motion of
the molecules. The partition function of a harmonic oscillator can be easily evaluated for both the
quantum and classical cases,[126] and from these expressions the free energy can be written as a sum
of contributions over all frequencies. Thus, classically,
βAharm =X
j
ln ~ωj
kBT+βACM +Nln srot,(4.4)
43
and for the quantum oscillator,
βAharm =X
j
ln 2 sinh ~ωj
2kBT+βACM
+Nln srot.(4.5)
where kBis Boltzmann’s constant, Tis the temperature, β= 1/kBT,~=h/2πwith hPlanck’s
constant; srot is the number of proper rotations of the molecule (2 for H2O), and ACM is the center-of-
mass correction to the free energy, which is given by [32]
βACM =−d
2ln N+ ln ρ, (4.6)
in which d= 3 is the number of spatial dimensions, and ρis the molecule number density. The
number of frequencies in the sum is 6N−3, which is the total dof of the system, minus those
corresponding to translation of the whole supercell in the three coordinate directions.
We now review the procedure for evaluation of the frequencies ωj. The process begins with
the equations of motion for the displacements ui
α(lκ), expressing the change in linear and angular
momenta in terms of the forces and torques that correspond to Eq. (4.3). The lattice symmetry dictates
a plane-wave solution, which when substituted into the equations of motion yields a determinantal
equation for the frequencies:[125, 124]
D(k)−ω2(k)M= 0,(4.7)
where D(k)is the 6n×6ndynamical matrix with elements
Dii′
αβ (k, κκ′) = X
l′
φii′
αβ (0κ;l′κ′) exp [ik·l′](4.8)
(note that iin the exponential here is √−1). The 6n×6nmatrix Min Eq. (4.7) has the molecule
masses and 3×3inertia tensors on its diagonal.[125, 124] The vector l′is the Cartesian position (with
respect to an origin at l= 0) of the lattice site indexed by l′. It is interesting to mention here that one
can reach the same classical free energy result using a different route that avoids computing Mwhile
diagonalizing the force constant matrix directly. Elsewhere[127] we demonstrate mathematically and
44
numerically the equivalence between the two approaches. However, this alternative approach cannot
be used in the quantum treatment; i.e. one must always solve Eq. (4.7).
The wave vectors kare those allowed from the first Brillouin zone of the supercell. That is, if
we have a supercell formed from nc1×nc2×nc3unit cells in the respective lattice-vector directions
(a1,a2,a3), then the allowable values of kare k= (m1/nc1)b1+ (m2/nc2)b2+ (m3/nc3)b3, where
(b1,b2,b3)are the reciprocal-lattice vectors of the unit cell, and where miare integers satisfying
−nci <2mi≤nci. Thus, for the 1×1×1supercell we have only k= 0; for 2×2×2we have eight
values of kformed from each of the mi={0,1}; for 3×3×3we have 27 values, mi={−1,0,1};
4×4×4we have 64 values, mi={−1,0,1,2};etc.
For each k, solution of Eq. (4.7) yields 6nvalues of the frequency ωi. We need not perform the
eigenvalue calculation for all allowable wave vectors, because we can exploit the Hermiticity of the
dynamical matrix, from which we know that ω(−k) = ω(k).[29] It should also be noted that the
calculation of the derivatives φii′
αβ (0κ;l′κ′)— each of which involves convergence of a lattice sum of
Lennard-Jones interactions and an Ewald sum — need be performed just once for each (αβκκ′ii′),
and in particular they do not have to be recomputed for each of the wave vectors. This economy
provides a significant computational advantage over an approach that includes the wave vector kin
the lattice and Ewald sums for φ.[125]
Additional detail regarding the lattice dynamics calculations may be found in the appendices.
4.2.2 Proton-Disorder Averaging
As mentioned above, the crystal structure of a clathrate hydrate specifies the positions of only the
oxygen atoms. The hydrogen (proton) positions are not specified, but their arrangement nominally
must satisfy the Bernal-Fowler ice rules.[128, 129] These rules state that each oxygen is covalently
bonded to two hydrogens, each oxygen forms two hydrogen bonds to two other oxygens, and there
exists exactly one hydrogen between a pair of neighboring oxygens (violations of these rules are
known as Bjerrum defects[130]). In addition, it is usually taken that the water molecules must orient
such that the structure as a whole has negligible dipole moment, because the bulk crystal has no
45
dipole.
Absent any defects, the proton-disordered isomers may be treated as independent components
of the overall partition function, such that the total free energy can be given as a sum over all
isomers.[129] If we use νto index the isomers, so that the free energy of the system constrained
to isomer νis A(ν), then the full free energy of the system is given by the sum over isomers:
βA =−ln X
ν
exp [−βA(ν)]
≈ −ln X
ν
exp [−β(U0(ν) + Aharm(ν))] (4.9)
where we separate Ainto its contributions from the static lattice energy and the harmonic vibrations,
to emphasize that each depends on the isomer ν. Topologically identical proton arrangements are each
represented in the sum; an alternative is to sum over distinct isomers, and multiply by a degeneracy
factor.
It is not a trivial task to evaluate the number of isomers that comply with the ice rules, but Pauling
estimated[131] the number of isomers as (3/2)N, and this has been shown to be accurate enough for
many purposes. Thus, the sH structure, which with 34 water molecules has the smallest unit cell, has
about one million isomers for the single unit cell. Many of these will have a non-negligible dipole
moment, and would be excluded on that basis, but this would not be known until they were generated
and examined. With such a number, consideration of all possible isomers is not feasible; so instead
we can sample them, using Msamples. Accordingly, we estimate the proton-disordered free energy
as
βA =−ln 1
M
M
X
ν
exp [−βA(ν)]!−Nln(3/2)
=βhU0+Aharmiexp −Nln(3/2) (4.10)
which uses Pauling’s value for the total number of terms in the sum represented by the sample; we
use h...iexp to represent the exponential average. In many cases, a single sample νiis used, typically
46
the one having the lowest energy and a negligible dipole. Effectively, the assumption is that the free
energy is the same for all terms in the sum over ν, and Eq. (4.9) becomes
βA =β[U0(νi) + Aharm (νi)] −Nln(3/2) (4.11)
It has been observed[129] that Aharm varies less than U0does across different isomers, leading to an
approximation in which U0is averaged while a single representative value is used for Aharm:
βA =−ln 1
M
M
X
ν
exp [−βU0(ν)]!+βAharm (νi)−Nln(3/2)
=βhU0iexp +βAharm (νi)−Nln(3/2) (4.12)
One also sees this approach employed but using instead a simple arithmetic average of U0:[118]
βA =1
M
M
X
ν
βU0(ν) + βAharm (νi)−Nln(3/2)
=βhU0i+βAharm (νi)−Nln(3/2) (4.13)
4.3 Computational details
Three different empty crystalline structures were considered, viz., sI, sII, and sH. sI is a cubic system
with P m3nspace group and contains 46 H2O molecules per unit cell; sII is a face-centered cubic
system with F d3mspace group and contains 136 H2O molecules per unit cell; and sH is a hexagonal
system with P6/mmm space group and contains 34 H2O molecules per unit cell. The lattice param-
eters of the sI and sII cubic systems used here are 12.03 ˚
A, and 17.31 ˚
A, respectively.[78, 122] We
combine two sH hexagonal unit cells to form an orthorhombic 68-molecule unit cell with dimensions
12.21 ˚
A×21.15 ˚
A×10.14 ˚
A.[122, 78] The coordinates of the oxygen atoms in the hydrate structures
are determined by X-ray diffraction;[90] in this work we used the oxygen positions give in Ref. [122]
as initial (unminimized) coordinates. The positions of protons (hydrogen atoms) are generated using
the algorithm described by Buch et al.,[132] which satisfies the ice rules.[128]
47
Table 4.1: TIP4P potential model parameters.[133]
rOH (˚
A) ∠HOH(deg) rOM(˚
A) A(kCal ˚
A12/mol) C(˚
A6/mol) qO(e) qH(e) qM(e)
0.9572 104.52 0.15 600.0×103610.0 0.0 0.52 -1.04
The TIP4P force-field model[133] was employed for the intermolecular interactions between the
H2O molecules. TIP4P is a rigid four-site model: three charged sites (on H1, H2, and M) and one
12-6 Lennard-Jones (LJ) site (on O). Table 4.1 shows the TIP4P parameters.
Lattice sums were evaluated for the LJ interactions between the oxygen atoms. The LJ cutoff
(Rcut) is set equal to 400 ˚
Afor both sI and sII and 500 ˚
Afor sH, to make sure that the truncation
error in the energy is less than 10−6kJ/mol. We employed a plain Ewald summation (ES) technique
[134] for the long-range Coulomb interactions. We set the ES real (rcut) and reciprocal (kcut) cutoffs
such that the truncation errors in the real and reciprocal energies are of the same order of magnitude
and less than 10−6kJ/mol based on the Kolafa et al.[135] formula for the truncation errors. Table 4.2
shows the specific cutoffs for different structures and supercell sizes used in this study.
The force-constant matrix for the local-harmonic approximation[136] is exactly the same as that
for the full harmonic analysis, except that only the 3×3block matrices (self linear and angular
interactions) along the diagonal of the full matrix are kept, while other interaction energies are set to
zero. Hence there is no need to compute any other terms to implement this approximation.
4.4 Results and Discussion
4.4.1 Structures generation and relaxation
Using Buch et al.’s algorithm,[132] 5000 proton-disordered structures were generated for each hy-
drate structure. Figure 4.1 shows the relationship between the potential energy and the net dipole
moment of those structures. The average values of the potential energy are -53.3301, -54.5052, and
48
Table 4.2: Real and reciprocal cutoff parameters of Ewald summation that give truncation error in the lattice
energy less than 10−6kJ/mol. ncis the number of unit cells in each direction forming the supercell, such that
Nc=n3
c.
sI
nc12345678
rcut(˚
A) 14.00 19.34 23.36 26.69 29.60 32.21 34.59 36.79
kcut(˚
A−1) 2.60 1.80 1.45 1.24 1.10 1.00 0.92 0.85
sII
nc12345
rcut(˚
A) 10.00 13.80 16.68 19.10 21.13
kcut(˚
A−1) 3.60 2.50 2.00 1.72 1.52
sH
nc1234567
rcut(˚
A)20.00 27.69 33.48 38.30 42.50 46.28 49.72
kcut(˚
A−1) 2.00 1.38 1.12 0.96 0.85 0.77 0.71
-53.5343 kJ/mol for sI, sII, and sH, respectively. The dipole moments fall into bands that are in agree-
ment with those computed by Takeuchi.[122] For each structure, 100 of the 5000 H-isomers with the
smallest dipole moment (regardless of energy) were chosen to compute the free energy.
For the purpose of harmonic analysis, all molecules must be at their respective minimum-energy
positions and orientations. We employed the steepest descent method to relax the molecular coor-
dinates for the 100 H-isomer structures of each hydrate. Several minimization steps (∼200) were
performed until the forces and torques on all molecules were of order of 10−4(in units of ˚
A, Dal-
ton, and ps). Among the 100 H-isomers, the structure with the minimum lattice energy was se-
lected for computing the free energy as will be shown below. Those structures have potential en-
49
0510 15 20 25 30
Dipole moment, P(2.18D)
-54.6
-54.4
-54.2
-54.0
-53.8
-53.6
-53.4
-53.2
-53.0
-52.8
Potential energy, U(kJ/mol)
CS-I
CS-H
CS-II
Figure 4.1: The potential energy versus the net dipole moment of the proton-disordered sI, sII, and sH hydrate
structures generated in this study.
ergies before (after) minimization of −54.068731 (−55.263099),−54.768947 (−55.400356), and
−54.175096 (−55.262875) kJ/mol.
4.4.2 H-isomer averaging
Since both the lattice and the harmonic free energies enter into the exponential averaging formula,
Eq. (4.9), the variation in both quantities affect the final free-energy results. We found that the
standard deviations of U0are ∼0.03,0.01,0.02 kJ/mol; and of Aharm are ∼2.31 ×10−5T, 7.06 ×
10−6T, 2.86 ×10−5TkJ/mol (where Tis in K) for sI, sII, and sH, respectively: the variation in U0
dominates over that of Aharm, and hence, it is reasonable to consider only one isomer for the harmonic
free energy and average over the potential energy (i.e., the approximation given by Eq. (4.12)).
Figure 4.2 shows the free-energy error (relative to the full exponential average of 100 isomers, Eq.
4.10) resulting from different approximations using the lowest-energy (solid line, i= 1) or second-
lowest-energy (dashed line, i= 2) H-isomers of 1×1×1sI hydrates. For the temperature range
studied here (0to 400 K), considering i= 1, exponential averaging of U0with a single-isomer Aharm
(magenta curve, Eq. 4.12) produced the smallest error (<0.025 kJ/mol), while using a single isomer
50
for U0(blue curve, Eq. 4.11) incurs larger error (∼0.07 kJ/mol for T > 100 K). The approximation
based on the arithmetic average of the lattice energy, denoted by h...i, (red curve, Eq. 4.13),[118]
produces large error at lower temperatures and approaches (as expected) the full exponential average
at higher temperatures. Comparison of the curves for the two isomers indicates that, given that the
selected isomer is among those having the lowest energy, the free energy is less sensitive to the choice
of isomer than to the choice of approximation.
4.4.3 Finite-size effects: thermodynamic limit
We perform extrapolation to the thermodynamic limit by computing free energies for supercells
of increasing size N, and examining the limit as 1/N →0. The supercell sizes used for each
clathrate structure are given in Table 4.2. In principle, the free energy computed for each super-
cell size should employ averaging over isomers, as described in the previous section. Moreover,
rather than use a supercell formed from replicas of a common proton-disordered unit cell, we should
use a “unit-supercell” in which proton disordering is performed across the entire structure. We will
use “exponential-average correction” (EAC) to refer to the difference between the free energy for the
proper isomer-averaged treatment, and that for the single (non-averaged) replicated-isomer supercell
used for the extrapolation. The computational expense involved in computing the EAC grows con-
siderably with N, which has led us to instead use an approximation. Specifically, we estimate the
free energy in the thermodynamic limit by extrapolating the free energy of supercells of increasing
size (each formed from the same H-isomer), and we apply to all system sizes (including the thermo-
dynamic limit) the EAC that is computing for the smallest system; that is, we assume the EAC is
independent of N.
We can gauge the error introduced by this approximation by computing the EAC for a 2×2×2
(nc= 2) supercell. The result is included in Fig. 4.2 as the green line labeled “2×2×2supercell”.
This difference should be compared to the EAC for the 1×1×1(nc= 1) supercell, given in the figure
by the blue solid line labeled “U0,i +Aharm,i”. The comparison shows that for Tabove about 150 K,
the EACs differ by about ∼10% of the EAC itself. The EAC is in turn only 25% as large as the
51
0 100 200 300 400
Temperature, T(K)
-0.10
-0.05
0.00
0.05
0.10
Free energy difference from exp. average, (kJ/mol)
i = 1
i = 2
U0,i + Aharm,i
< U 0 > + A harm,i
<U 0>exp + Aharm,i
< U 0 + Aharm > exp
2×2×2 supercell
Figure 4.2: The 1×1×1sI harmonic free-energy using various approximations (viz. Eq. 4.11 (blue), Eq.
4.12 (magenta), and Eq. 4.13 (red)) expressed as a difference from the full exponential average (Eq. 4.10).
Approximations using two isomer choices are shown: the i= 1 H-isomer (solid lines) is the structure with the
minimum lattice energy, and i= 2 is the second-lowest one (dashed lines). The green line (labeled 2×2×2)
is the same as the blue solid line but using 2×2×2supercell and with the single isomer being a replica of the
i= 1 unit cell.
finite-size effect in going from the nc= 1 to the nc= 2 supercell (described below). Another factor
to consider is the use of a finite number of samples (100) in computing the exponential averages. This
leads to an uncertainty of about 5% in the nc= 1 EAC (blue curve) (∼0.007 and 0.003 kJ/mol for
T= 100 and 300 K, respectively), which may account for some of the difference with the nc= 2
EAC. In sum, the results suggest that for T > 150 K, it is reasonable to use the nc= 1 EAC as a
surrogate for the corresponding correction at each system size, including the thermodynamic limit.
The summation over ln ωiin Eq. (4.4) is the only non-trivial N-dependent term of the classical
harmonic free energy. According to Hoover,[137] the leading Ndependence of this summation should
be ln N/N, so in subtracting this contribution from the sum in Eq. (4.4) we should produce a system-
size dependence that is closer to linear in 1/N. Hoover’s conclusion was based on observations of
the analytic behavior for a 1D harmonic system, and numerical results for systems in 2D and 3D,
both involving only translational degrees of freedom. Figure 4.3 confirms that his analysis is correct
52
00.0005 0.001 0.0015 0.002 0.0025 0.003
1/N
34.16
34.20
34.24
34.28
34.32
Σiln(ωi)/N - C ln(N)/N
C = 0
C = 1
34.2205 - 13.3300/N
34.2617 - 13.4400/N
34.3172 - 13.9343/N
sH
sII
sI
Figure 4.3: Finite-size effects on the classical harmonic free energy for the sI, sII, and sH structures. The filled
points are harmonic free-energy contributions with the leading ln N/N term subtracted (C= 1), and the open
circles are the raw data (C= 0). Dashed and solid lines are linear fits (in 1/N) to the C= 0 and C= 1 cases,
respectively. Note that the results for the smallest Nfor each structure are not shown, as they are off the scale
to the right; however all points are included in the linear fits.
also for molecular systems, in that the data are clearly more linear in 1/N when ln N/N is subtracted
(C= 1). This suggests that the general trend observed by Hoover, viz., that the ln N/N leading term
of the free energy is independent of the number of dof per molecule, persists even with rotational
motions included. In addition, although the extrapolation to infinite system size (1/N →0) yields a
different intercept for each structure, the slopes are nearly equal (differing by from each other by less
than 0.5%).
For the quantum system, unlike the classical treatment, the extrapolation to infinite system-size
depends on the temperature (see Eq. (4.5)). Moreover, since the quantum and classical behavior are
the same only for high temperatures, we can count on ln N/N to be the leading term only at those
conditions. We performed fits of the harmonic free energy with the ln N/N contribution removed
(as with the classical case), regressing a new curve for each temperature. We find that the linear fit
is inadequate at lower temperatures, but that a quadratic form (in 1/N) was effective in describing
the data. The fit matches the data to within 10−6kJ/mol, which is of same order as the convergence
53
100 150 200 250 300
Temperature, T (K)
-49
-48.5
-48
-47.5
-47
Free energy, u0 + aharm (kJ/mol)
1×1×1
N → ∞
III
H
I
II
H
classical
Figure 4.4: Classical harmonic Helmholtz free energy, per unit molecule, for the sI, sII, and sH structures as a
function of temperature in the thermodynamic limit (solid lines) and for a single unit cell (dashed lines). The
exponential-averaging correction is included in all curves.
tolerance in the lattice and Ewald sums. We do not show plots of the fits, and instead just present the
N→ ∞ values in the next section.
4.4.4 Free-energy results
Figures 4.4 (classical) and 4.5 (quantum) show the harmonic Helmholtz free energies (including the
EAC correction) in the thermodynamic limit versus temperature. In the whole temperature range
(100 to 300 K), sII has the lowest free energy according to both treatments; however, it is almost
indistinguishable from sI according to the quantum treatment while, classically, the difference is
noticeable. The systems are not all at the same density (differing from each other by about 1%), so
we cannot draw conclusions about relative stability from these data, and such conclusions are not
the goals of this study. Regardless, it is known that the empty lattices are all unstable relative to ice
phases.[138, 139]
The nearly-equal slopes of the harmonic free energy with 1/N (cf. Fig. 4.3) suggest that finite-size
effects impact all crystal structures nearly equally, so if the interest is only in computing differences
54
100 150 200 250 300
Temperature, T (K)
-46
-45
-44
-43
-42
-41
Free energy, u0 + aharm (kJ/mol)
1×1×1
N → ∞
III H
I
II
H
quantum
Figure 4.5: Quantum harmonic Helmholtz free energy, per unit molecule, for the sI, sII, and sH structures as a
function of temperature in the thermodynamic limit (solid lines) and for a single unit cell (dashed lines). The
exponential-averaging correction is included in all curves.
between the structures, then a small system would suffice. However, in many studies the comparison
is made for convenience between systems of the same number of unit cells Nc, not the same number
of molecules N. To demonstrate the effect of comparing free energies in this manner, Figures 4.4
and 4.5 show free energy curves based on calculation for a single unit cell. In both the quantum and
classical treatments, differences in free energy between structures decrease markedly when using only
one unit cell, and the effect is strong enough to affect the ordering of the curves. For the quantum
treatment, the free energies of the sI and sII structures when each are described by a single unit cell
are indistinguishable around 100 K before they become different for high temperature; for the infinite
system, their relative behavior is in fact the opposite of this trend.
Our single unit cell results show a qualitative agreement with that of Tanaka et al.[139, 138] for
empty sI and sII structures. We do not expect an exact agreement, for several reasons: they averaged
(in an unspecified manner) their results over only 6proton-disordered structures; we used a different
technique to calculate the long-range Coulomb interactions, and employed much larger cutoffs for the
dispersion interactions; also, they did not include the srot contribution (Eqs. (4.4) and (4.5)).
55
0 200 400 600 800 1000
Frequency, ω (cm-1)
0
0.002
0.004
0.006
0.008
0.01
Normalized phonon density of states
sI: 8×8×8
sII: 5×5×5
sH: 7×7×7
Figure 4.6: Normalized phonon density-of-states (PDOS) of the largest supercells considered in this work for
sI, sII, and sH structures. sI and sII are shifted in y-axis by 0.006 and 0.003, respectively, from sH to avoid
overlap.
4.4.5 Phonon density-of-states
Figure 4.6 shows the phonon density-of-states (PDOS) for the largest system sizes studied in this
work. The high-resolution, high-precision histogram was built using at least 100,000 normal modes
for each of the three empty clathrate structures, with narrow-width (σ= 4 cm−1) Gaussian smearing
used to smooth the data. One common feature of all structures is the existence of a large gap separating
two different bands. Based on eigenvector arguments, the lower and upper bands are mostly for
translation and rotation modes, respectively.[114, 99, 100] Our PDOS results show good agreement
with previous work. [114, 112, 139, 38, 99, 100, 140]
4.4.6 Accuracy of the local-harmonic approximation
Although results for the local-harmonic approximation (LHA) were reported previously for different
clathrate hydrates,[36, 37, 39] comparison with the full harmonic free energy was not performed.
Figure 4.7 shows such a comparison for our three structures in the thermodynamic limit. In this
temperature range, the absolute free energies differ by at least 0.8kJ/mol. Although this difference
56
100 150 200 250 300
Temperature, T (K)
-49
-48
-47
-46
Free energy, u0 + aharm (kJ/mol)
LHA
Full
III H
III H
Figure 4.7: Relative stability based on classical free energies of single H-isomer (the minimum), in the thermo-
dynamic limit, of sI, sII, and sH structures using full and local harmonic (LHA) methods.
increases with the temperature, the relative ordering of the structures is preserved.
4.5 Conclusions
We considered lattice dynamics methods for calculation of the free energy of clathrate hydrate phases,
specifically the cubic structure I (sI), cubic structure II (sII) and hexagonal structure H (sH) phases, in
the absence of guest molecules; water molecules are modeled with the TIP4P potential. The analysis
performed here aims to demonstrate the significance of finite-size effects and proton disorder when
performing lattice dynamics calculations to estimate the free energy of clathrate structures. Although
the development involves the evaluation and comparison of free energies for different structures, we
do not mean for these results to be interpreted in regard to the relative stability of the structures, nor
to identify a phase transition. We do not, for example, include guest molecules at all, and we do not
attempt to ensure that the systems are at a common pressure or even at the same density. Instead, the
comparisons are made to provide a sense the variation of the free energy between structures, so as to
provide a scale for gauging the magnitude of the effects under study.
57
Within this context, the results demonstrate that the effects involved can be significant. We can
summarize some of the key considerations (some of which are already known and thus were not
detailed above), keeping in mind that the scale of the difference in free energies between the empty
structures is at most 0.2 kJ/mol.
•Finite size effects (one unit cell versus the infinite system) are, at 300 K, in the range of 0.15
to 0.5 kJ/mol, depending on crystal structure. Of particular importance is the need to compare
systems having equal (or nearly equal) numbers of molecules, rather than comparing systems
having the same number of unit cells (which often is only a single cell). The error in the free-
energy difference between structures incurred by using a single unit cell can be of order 0.4
kJ/mol.
•Proton disorder (as measured by the exponential-averaging correction) affects the free energy
by about 0.03 to 0.06 kJ/mol, again depending on crystal structure.
•Quantum effects are of the order of 5 kJ/mol, and differ between structures by about 0.1 kJ/mol.
•Point defects, including vacancies, interstitials, and Bjerrum defects are stable at equilibrium,
and can affect the free energy. We do not have data regarding defects in clathrates, but calcula-
tions have been performed for ice structures (e.g., Ref. [141]).
We also wish to point out the usefulness of the well-established methodology for analytic evalu-
ation of orientational derivatives of rigid molecules, which can be used to avoid numerical differen-
tiation; the relevant equations are summarized in Appendix A. We mention this because, while some
researchers do perform derivative evaluations in the same manner that we have, it is also not unusual
to encounter studies that use numerical evaluation of derivatives. Such an approach can be compu-
tationally expensive when also doing Ewald and lattice sums. In this regard, our formulation for the
elements of the dynamical matrix, Eq. (4.8), is particularly efficient when studying systems requiring
Ewald summation. Also of note is the observation that the leading-order finite-size correction for
classical molecular systems is −ln N/N, as is known for monatomic molecules.
58
It would be of interest to extend these studies to put them in the context of occupied clathrate
systems. It would also be worthwhile to conduct a similar study for the various phases of ice. With
the development of very efficient methods for evaluation of anharmonic contributions to the free
energy,[33, 34] we may soon find that proton disorder and finite-size effects will prove to be the
limiting factor in obtaining accurate free energies. These effects can be efficiently handled with
lattice-dynamics methods, which ultimately will be used as a key part of an analytical + simulation
strategy for the calculation of solid-state phase diagrams from first principles.
59
CHAPTER 5
Direct Measurement of Anharmonic Properties of Solids
5.1 Introduction and Background
As mentioned in Chapter 2, the lattice dynamics (LD) method forms the foundation for our under-
standing of the properties of crystalline systems [123, 28]. It is based on the assumption that the
intermolecular energy of the crystal is, for the given volume, at a local minimum with respect to atom
positions, and that lattice vibrations are small in magnitude compared to the intermolecular spacing.
This permits the use of a harmonic approximation, in which the energy is expanded to second order
in the atom displacements. The result is an approximate Hamiltonian that can be solved in closed
form for the dynamic and thermodynamic behaviors, using either quantum or classical mechanics.
Volume-dependent thermal effects can be evaluated with the quasi-harmonic extension of this treat-
ment.
At low temperature or high pressure, LD works very well, and provides a description of crys-
tal behavior that is adequate for many purposes. However, atomic vibrations grow with increasing
temperature and decreasing density, and the harmonic approximation begins to fail. Consequently,
there are many conditions of interest for which LD is inadequate. In such situations, the most reliable
alternative is molecular simulation.
Molecular simulation as normally practiced for the evaluation of anharmonic thermal properties
does not exploit the harmonic character of the crystal to improve calculation of averages. Thus,
even at conditions where LD provides an excellent description, simulation essentially “starts from
scratch” in evaluating the properties, making no use of the LD characterization. Consequently, the
precision of the properties computed by molecular simulation is severely compromised, inasmuch as
the stochastic averaging must contend with fluctuations contributed by the harmonic component of the
intermolecular potential. There is a clear inefficiency in computing stochastically a large contribution
that is already known analytically.
In this chapter, we outline and demonstrate an approach that remedies this problem. By repur-
posing a class of methods formulated to improve free-energy (FE) calculations, we show that the
harmonic character of the crystal can be leveraged to provide in essence a direct measurement of the
anharmonic contributions to the properties. See Sec. 5.2 for detailed derivations and formulas.
5.2 Harmonically-Mapped Averaging (HMA)
Here, we are repurposing a class of methods formulated to improve FE calculations (due to Jarzynski
[142]). The harmonic character of the crystal is leveraged to provide in essence a direct (and efficient)
measurement of the anharmonic contributions to the properties.
5.2.1 General Formulation
Assume a coordinate mapping r0→r(r0, λ)couples well-defined collective molecular displace-
ments to a FE perturbation trial, λ0→λ, such that the positions rare more appropriate than r0
to the perturbed state λ. Although intended for use with a finite perturbation, Jarzynski also pre-
sented a differential form, which we reproduce and extend here. Then, rather than use this toward
61
FE calculations, we employ it instead to devise new forms of ensemble averages for thermodynamic
properties. The resulting framework provides the mechanism needed to introduce efficiencies based
on the harmonic character of the crystal; the key is the selection of an appropriate mapping.
The FE is a function of the perturbation parameter λ(which we treat as multivariate); thus in order
to get FE derivatives let’s first Taylor expand ∆Ain λto get
∆A=Aλ·∆λ+1
2∆λ· Aλλ ·∆λ+... (5.1)
where Aλand Aλλ are first and second derivative of Awith respect to components of λand evaluated
at λ=λ0. The functional form of those derivatives will depend on the formula used to get ∆A. In
our case, we will be using the targeted free energy perturbation formula due to Jarzynski [142]. He
showed that ∆Acan be expressed as an ensemble average in the λ0system:
∆A=−ln Je−∆Uλ0,(5.2)
where ∆U ≡ U (r;λ)− U (r0;λ0)is the configurational energy difference (in units of kBT), and
J(r0;λ)≡ |∂r/∂r0|is the Jacobian of the mapping. In a similar fashion both Jand ∆Uwill be
Taylor expanded
J= 1 + Jλ·∆λ+1
2∆λ·Jλλ ·∆λ+... (5.3)
and
∆U=Uλ·∆λ+1
2∆λ· Uλλ ·∆λ+... (5.4)
Note that the first term in the Jexpansion is 1due to the fact that J(λ0→λ0,r0) = 1 and r=r0
at ∆λ= 0. Plugging Eqs. 5.3 and 5.4 into Eq. 5.2 (using e−x=−x+1
2x2+... and ln(1 + x) =
x−1
2x2+...) and keeping the first and second coefficients of ∆λwe get the following when compared
to Eq. 5.1
Aν=−hJνi+h Uνi(5.5)
Aµν =hJµihJνi − hJµν i+h Uµν i(5.6)
+ Cov [ Uµ, Jν] + Cov [Jµ,Uν]−Cov [ Uµ,Uν],
62
where the subscripts νand µare components of λand Cov [X,Y] = hXYi−hXihYi. All derivatives
and ensemble averages are evaluated at λ=λ0. In the case where Jis independent of r0, then
Aν=−Jν+h Uνi(5.7)
Aµν =JµJν−Jµν +h Uµν i − Cov [ Uµ,Uν],(5.8)
for which there is a clear separation into mapping (involving J) and residual (involving U) compo-
nents.
The derivatives of U(r(r0, λ); λ)can be separated into a part due to the mapping, and a part due
to the direct dependence on λ. Using ∂νfor the νderivative taken with rfixed at r0,
Uν=∂νU − βF·rν(5.9)
Uµν =∂µν U − βF·rµν +βrµ·Φ·rν(5.10)
−[∂µ(βF)·rν+∂ν(βF)·rµ],
where βF≡ −∇r0Uis the force vector and βΦ≡ ∇r0∇r0Uis the force-constant matrix; all deriva-
tives are evaluated at r=r0and λ=λ0. Once the mapping is specified, the approach is easily
implemented, and entails only a simple reformulation of the averages taken in an otherwise standard
molecular simulation.
5.2.2 Application: Tand Vfree-energy derivatives
Molecular simulation as normally practiced for the evaluation of anharmonic thermal properties does
not exploit the harmonic character of the crystal to improve calculation of averages. Thus, even at
conditions where LD provides an excellent description, simulation essentially “starts from scratch” in
evaluating the properties, making no use of the LD characterization. Consequently, the precision of
the properties computed by molecular simulation is severely compromised, inasmuch as the stochastic
averaging must contend with fluctuations contributed by the harmonic component of the intermolec-
ular potential. There is a clear inefficiency in computing stochastically a large contribution that is
already known analytically.
63
Here, we consider the specific case in which the FE derivatives are with respect to temperature
and volume, i.e., λ≡(T, V ). We will then arrive at expressions for the energy and pressure (via first
derivatives), heat capacity, bulk modulus, and thermal pressure coefficient (via second derivatives).
Before that, however, it is useful (see below) to introduce a new coordinate system consistent with
the volume change. Normally, the volume derivatives of Uare done through uniformly scaling the
atomic coordinates — the first derivative, for example, gives the well-known virial pressure. Since
we are not using uniform volume-scaling (see Eq. (5.11)), it is more convenient to choose coordinates
that can, naturally, factorize those derivaties into uniform (what is normally done in literature) and
off-uniform contributions. This can be achieved using fractional coordinates in which the initial and
scaled positions are divided by their respective box length (L0and L, respectively). For convenience,
r(and r0) will be used to denote the fractional coordinates. In this system, the potential energy can
be written as U(Lr;λ); where the dependence on the absolute positions is given in terms of Land
r, separately. Thus, for example, the first volume-derivative, ∂VU, is interpreted as how much the
energy changes with volume at fixed fractional coordinates, r. This is nothing but the uniform scaling
usually used to get the virial pressure; same applies for the second derivatives.
Now, we introduce the following mapping, appropriate to a classical monatomic crystalline sys-
tem. In this transformation, the mapped fractional deviation from the lattice site Ris given by
∆r=f(β, V ) ∆r0,(5.11)
where ∆r≡r−R,∆r0≡r0−R0(however, R0=Rin our fractional system), and the function
f(β, V )is defined as
f(β, V )≡L0
Lβ0
β1/2V0
Ve−β0[δA−δUlat ]1/d(N−1)
(5.12)
where δX ≡X(V)−X(V0)is the change in property Xresulting from this scaling; Ulat (V)is the
perfect-lattice energy. The temperature dependence of this scaling is based on the T1/2dependence of
the vibration amplitude of a classical harmonic system; this scaling was used previously to formulate
an efficient temperature-perturbation scheme to measure the FE of crystals [33, 34]. For the volume
dependence, we estimate δA ≈ − ˆ
P δV +ˆ
BδV 2/2V, where ˆ
Pand ˆ
Bare parameters selected to
64
approximate the pressure and bulk modulus, respectively. It is important to emphasize here that
although the measured properties do not depend on those parameters, the uncertainties do — there is
always an optimum value that minimizes the uncertainty. For reasons related to finite-size effects (see
Chapter 5) we used quasi-harmonic analysis to estimate them; however, we observed that this choice
is not far from the optimum values. A similar scaling was used recently to formulate an efficient
volume-change move for isobaric simulations of crystals [143]. Both the volume and temperature
scaling can be reasoned as distributing the non-lattice free-energy change of the crystal uniformly
and isotropically among all the molecules.
Given this mapping, the corresponding Jacobian of transformation and energy derivatives needed
to evaluate Eqs. (5.5) and (5.7) can be easily derived. First, the Jacobian (now is given by |∂∆r/∂∆r0|
because Ris constant) can be shown to be
J(β, V ) = β0
βd(N−1)/2L0V0
L V e−β0(δA−δUlat).(5.13)
This is independent of r0, so we can use the form of the derivatives given in Eqs. (5.7) and (5.8).
Second, the energy derivatives can be obtained by plugging the first and second derivatives of r(viz.
rν=fν∆r0and rµν =fµν ∆r0) into Eqs. (5.9) and (5.10) yields
Uν=∂νU − βfνF·∆r(5.14)
Uµν =∂µν U − βfµν F·∆r+βfνfµ∆r·Φ·∆r
−[fν∂µ(βF) + fµ∂ν(βF)] ·∆r.(5.15)
Note that for simplicity we dropped the the subscript 0here (and for the rest of the paper); however,
it does not matter as L0cancels out due to multiplication. An interesting feature of Eqs. (5.14) and
(5.15) is that, in the case of volume derivatives, the off-uniform quantities (i.e. other than ∂VUand
∂µV U) are given in terms of ∆rand the total forces (and their derivatives); thus, periodic-boundary
condition complications of the uniform quantities (Pvir and ∂VPvir; see Table 5.1) are not relevant if
the task is to compute only the off-uniform component. This is especially a useful formulation when
commercial packages are used (e.g. LAMMPS and VASP) in which the uniform quantities are given
and only the total forces (and their derivatives) can be extracted.
65
Table 5.1: Ensemble averages required for first- and second-derivative properties.
Property Definition Conventional Average Harmonically Mapped Average
Configurational
Energy
U=AβhUid(N−1)
2β+U+1
2F·∆r
Pressure P=−AV
β
N
V β +hPviri∆ˆ
Pharm +hPvir +C1F·∆ri
Isochoric
Heat Capacity
T Cv=−βAββ βVar [U]d(N−1)
2β+βVar [Uanh]−1
4hF·∆r+ ∆r·Φ·∆ri
Isothermal
Bulk Modulus
B
V=AV V
β
N
V2β− h∂VPviri − βVar [Pvir]
1
V∆ˆ
Bharm −βVar [Panh]− h∂VPvir +C2F·∆r
+2C1∂VF·∆r−C2
1∆r·Φ·∆r
Isochoric Thermal
Pressure Coefficient
T γv=AβV −AV
β
N
V β +βCov [U, Pvir]
∆ˆ
Pharm +βCov [Uanh, Panh] + 1
2hC3F·∆r
+∂VF·∆r−C1∆r·Φ·∆ri
Uanh ≡U−Ulat +1
2F·∆r;Panh ≡Pvir +C1F·∆r−Plat ;Pvir =−∂VU;Plat =−∂VUlat ;Blat =−V ∂VPlat;ρ≡N/V ;
C1≡β∆ˆ
Pharm−ρ
d(N−1) ;C2≡C2
3+V−2−βV −1∆ˆ
Bharm
d(N−1) +d−1
(dV )2;C3≡C1+1
dV ;∆ˆ
Pharm ≡ˆ
Pharm −Plat ;∆ˆ
Bharm ≡ˆ
Bharm −Blat
Expressions for some thermodynamic properties derived from these formulas (using harmonic
estimate of ˆ
Pand ˆ
B) along with the conventional counterpart are summarized in Table 5.1. To
illustrate that these “harmonically mapped averages” give directly the anharmonic contribution, let
us consider the expression for the energy, U. For a harmonic system, −1/2F·∆ris, in every
configuration, exactly equal to the total energy (beyond Ulat ), so the mapped average (U+1/2F·∆r)
is identically zero if applied to a harmonic system. Added to this average is the analytic expression
for the energy of a harmonic system (1/2kBTper each degree of freedom), which emerges naturally
from the mapping contribution in Eq. (5.7). The volume-perturbation mapping is less rigorous—it
assumes all phonon modes scale equally with volume—so for a harmonic system we cannot expect a
contribution of zero to the volume-derivative averages for every configuration. For the pressure, the
primary contribution is ˆ
P, and added to this is an average comprising terms that tend to offset to the
extent that the system is harmonic.
66
5.3 Application to LJ potential
5.3.1 Simulation Details
We demonstrate with application to the Lennard-Jones (LJ) model with both σand ǫare set to unity.
We performed standard canonical-ensemble (N V T ) Monte Carlo (MC) simulations over thermody-
namic conditions within the region of stability of the fcc solid [144]. The performance over the
conventional method is presented using cubic box of N= 500 atoms and bigger systems (4n3atoms,
with n= 6,7, ..., and 10) to study finite-size effects. The LJ potential is truncated at rc= 3.0every-
where except for the long-range correction analysis at which we used bigger cutoffs (3.5,4.0, ..., and
7.5). We collected both conventional and mapped averages as given in Table 5.1. The quasi-harmonic
estimate of the pressure and bulk modulus ( ˆ
Pharm and ˆ
Bharm) are obtained through polynomial fitting
of the harmonic free-energy versus volume and then differentiate, analytically, the fitting function.
Runs of 107MC trials were performed (after 2×106steps of equilibaration) at each state point, and
each run was repeated 100 times to collect error statistics. We also conduced canonical-ensemble
molecular dynamics (MD) simulation to study the effect of time-step on the averages. The temper-
ature was controlled using Andersen thermostat every 100 steps. In this study we follow (unless
otherwise mentioned) an isochore of density ρ= 1.0and an isotherm of temperature T= 1.0. Ac-
cording to [144], the melting temperature of that isochore is Tmelt = 0.930 while the solid melting
density of the isotherm is ρmelt = 1.011.
CPU used to benchmark the code
5.3.2 Difficulty of Measuring Free-energy Derivatives
Figure 5.1 provides a direct picture of the fluctuations in calculation of the energy. The comparison
shows clearly that the conventional method suffers greatly from fluctuations in the harmonic contri-
bution to the integrand. What is particularly notable in this case is how small the mapped-average
fluctuations are relative to the difference from the harmonic-contribution average. The mapped aver-
67
age almost immediately provides a value of the anharmonic contribution that is statistically significant
on this scale. In contrast, the conventional method would require a long period of averaging to be able
to discern this difference, i.e., to provide an energy that is meaningfully different from the simple har-
monic energy.
Figure 5.2 plots the average energy for two isochores, each from near-zero temperature up to the
melting temperature for the given density (Tmelt = 2.615 at ρ= 1.2[144]). The data are presented
as Uanh/T 2, which remains finite as T→0; this is the integrand used to compute FE differences via
thermodynamic integration (TI) in temperature along an isochore (see Sec. 5.3.3). The size of the
error bars and the scatter in the data clearly show (especially with the denser system) that the result
from the mapped average is much more precise, for the same amount of computation. Consequently,
the precision of FE estimation from TI is improved (see Sec. 5.3.3).
Figure 5.3 illustrates the same, but for the isochoric heat capacity CV, a second-derivative property.
The trivial 3(N−1)kB/2harmonic contribution is subtracted and the data is presented as CV,anh/T ,
which is finite as T→0. It can be noticed that the conventional data is largely scattered around the
average; this is because CVis given as a variance of the conventional energy (see Table 5.1), which is
noisy by itself as shown in Fig. 5.1. On the other hand, the mapped average data shows less scattering
because CV,anh is given as a variance of the less noisy anharmonic energy plus a canonical average
that vanishes near-zero temperature (see Table 5.1).
Figures 5.4-5.6 show the performance of the methods in terms of the “difficulty” (D≡t1/2σ)
of measuring some quantity in time twith uncertainty σ. This measure accounts for any extra com-
putational effort needed to perform the mapped average, providing a more fair comparison with the
conventional method than might be given by the uncertainties alone [145]. Each plot compares per-
formance via the ratio of difficulties of the methods (Dconv/Dmap). The square of this quantity gives
the ratio of CPU-times required by conventional versus mapped averaging to achieve a result of the
same precision. Performance improvements appear to cluster according to whether the property is a
first- or second-derivative quantity.
Figures 1(d) and (e), respectively, demonstrate the temperature and volume dependence of the
68
0 2000 4000 6000 8000 10000
MC steps
0.24
0.26
0.28
0.3
0.32
0.34
(U-Ulat)/N (LJ units)
Conventional
Harmonically mapped
Harmonic-contribution average
(a) ρ = 1.0 T = 0.2
Figure 5.1: (Color online). Conventional and HMA potential energy (off lattice contribution) trajectories of a
500 atoms arranged in FCC structure and interacting through LJ potential.
performance. In each case, the system becomes more harmonic toward the zero of the abscissa, and
reaches melting at the opposite end. As the conditions move away from melting, the performance
improvement shown by harmonic mapping becomes just extraordinary. First-derivative properties are
computed with 100 to 10,000 times less effort, while speedup for the second-derivative properties
exceeds a millionfold for the highest density examined.
Figure 1(d) gives the difficulty ratio for each property along the melting line, starting from the
triple point (Ttp = 0.687 and ρtp = 0.963) [144]. These are the conditions where the crystal deviates
most from harmonic behavior, and thus where harmonically-mapped averaging is least effective. Even
here, performance of the new method is almost 20 times faster for first-derivative properties, and about
100 times faster for second-derivative properties.
The above second-derivative quantities were computed in a single simulation using fluctuation-
based formulas (see Table 5.1). However, an alternative route is to fit and then differentiate the
relevant first-derivative quantity over a whole range of thermodynamic states. Thus, comparing the
two schemes is only computationally fair if the interest is over that range (as we assume here) not just
in a single state. In addition, it is interesting to see how the efficiency of measuring first-derivative
quantities (conventional and mapping) will be reflected in measuring the second-derivatives.
69
0 0.2 0.4 0.6 0.8 1
T/Tmelt
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
Uanh/NT2 (LJ units)
Conventional
Harmonically mapped
(b)
ρ = 1.2
ρ = 1.0
Figure 5.2: (Color online). Conventional and HMA estimations of the specific anharmonic energy, along two
different isochores, from near-zero temperature to melting state. System as described in Fig. 5.1.
0 0.2 0.4 0.6 0.8 1
T/Tmelt
-1.0
-0.8
-0.6
-0.4
-0.2
Cv,anh/NT (LJ units)
Conventional
Harmonically mapped
(c) ρ = 1.0
Figure 5.3: (Color online). Isochoric anharmonic heat capacity along unity isochore, from near-zero tempera-
ture to melting state. System as described in Fig. 5.1.
70
0 0.2 0.4 0.6 0.8 1
T/Tmelt
1
10
100
Difficulty ratio, Dconv/ Dmap
U
P
Cv
B
γv
(d) ρ = 1.0
Figure 5.4: (Color online). Difficulty ratio (conventional/mapped) of different thermodynamic properties along
unity isochore from near-zero up to melting state. System as described in Fig. 5.1.
0.2 0.4 0.6 0.8 1
V/Vmelt
100
101
102
103
104
Difficulty ratio, Dconv/Dmap
U
P
Cv
B
γv
(e) T = 1.0
Figure 5.5: (Color online). Difficulty ratio (conventional/mapped) of different thermodynamic properties along
unity isotherm up to melting state. System as described in Fig. 5.1.
71
1.0 1.5 2.0 2.5 3.0 3.5
Density (LJ units)
2
4
6
8
10
12
14
16
Difficulty ratio at Tmelt, Dconv/Dmap
U
P
Cv
B
γv
(f)
Figure 5.6: (Color online). Difficulty ratio (conventional/mapped) of different thermodynamic properties along
the coexistence line. System as described in Fig. 5.1.
We will take the isochoric heat capacity CVas an example (the bulk modulus B(not shown here)
behaved similarly). The smoothness of Uanh/T 2data (see Fig. 5.2) makes it easier function to fit
rather than fitting the Uanh directly. However, the slope of that function is not trivially related to CV,
hence an extra analysis is needed to extract the uncertainty in the heat capacity from the raw data. We
used polynomial fit of second and forth orders for the conventional and mapped averages, respectively,
from near-zero temperature to 1.1Tmelt (to improve fitting results up to the melting point). Figure 5.7
shows the error in CV/T from fitting against (direct) measurement during simulation. It is clear
that the fitting is more efficient than the direct method for both conventional and mapped averages.
In addition, the direct measurement of CVusing the mapped method is comparable in efficiency to
the fitting the conventional energy. Nevertheless, fitting the mapped energy still wins against others;
hence, given the simplicity of the anharmonic energy formula, this should be the method of choice if
the focus is over a range of temperatures. However, if interest is only in a single point, then, the direct
mapped average should always be used.
72
0 0.2 0.4 0.6 0.8 1
T/Tmelt
0.001
0.01
0.1
1
Error in Cv/T (LJ units)
Direct
Fit Uanh/T2
ρ = 1.0
Conventional
Harmonically-mapped
Figure 5.7: (Color online) Comparison of the uncertainties from the conventional and mapped measurement of
heat capacity using “direct” method (i.e. using formulas in Table 5.1) and a fourth-order polynomial fit of the
energy.
0 0.2 0.4 0.6 0.8 1
T/Tmelt
0.01
0.1
Free energy difficulty (LJ units)
Frenkel-Ladd integration
Harmonically-mapped TI
Conventional TI
ρ = 1.0
Figure 5.8: (Color online) Difficulty in measuring the free energy. The small jumps in the TI results are due to
changing the order of the polynomial fit (see text).
73
5.3.3 Difficulty of Measuring Absolute Free-Energy
One of the important applications of measuring the energy is to estimate the FE using the well-
known TI technique [1]. Since our focus is on the stochastic uncertainty in the FE more than its
average (although we check for accuracy), we will measure only the anharmonic contribution (lattice
and harmonic contributions are deterministic) of the FE, Aanh. For a system at temperature T1, this
contribution is given by [32]:
βAanh (T1) = −ZT1
0Uanh
T2fit
dT. (5.16)
The integration is evaluated, analytically, over a fitting function of Uanh/T 2data from T= 0 to
some desired temperature T1(few more points are added to improve the fitted result at T1). We used
polynomial fit of increasing order (based on T1and the method of measuring Uanh ) to evaluate the
integrand of Eq. (5.16).
Figure 5.8 shows the difficulty results of measuring the FE using the TI method (both conventional
and mapped) along with results from the widely-used Frenkel-Ladd (FL) integration scheme [17].
The small discontinuity in the TI results is due to changing the polynomial order with temperature.
It can be noticed that while the FL difficulty is almost independent of T(because the integration is
done over internal coupling parameter) the difficulty in TI increases with temperature. Although the
computational cost of the conventional TI is comparable to the FL method, the harmonically-mapped
TI is about 50 −1000 faster (depending on T) than the standard FL method.
5.3.4 Thermodynamic Limit
Extrapolating properties to the thermodynamic limit (1/N →0and 1/rc→0) is needed to gauge
the behavior at the macroscopic level. In this section we estimate both the finite-size and long-range
corrections (FSC and LRC, respectively) exploiting the knowledge of the harmonic behavior in that
limit. Thus, the harmonically-mapped averaging is used as it estimates, directly and precisely, the
anharmonic contribution. The pressure is taken as an example to demonstrate these effects.
74
0 0.001 0.002 0.003 0.004
1/N
2.55
2.56
2.57
2.58
2.59
2.60
2.61
Pressure (LJ units)
0 0.001 0.002 0.003 0.004
-6.0
-4.0
-2.0
0.0
HarmFSC: 2.6054(9) - 1.3(5)/N
Direct: 2.6054(9) - 8.1(5)/N
ρ = 1.0 , T = 0.8
T = 0.2
T = 0.4
T = 0.8
HarmFSC
N = 256
N = 4000
×10-3 P - PN→ ∞
Figure 5.9: (Color online) Finite-size effects of measuring the pressure directly (no correction) and using the
HarmFSC ≡Pharm(∞)−Pharm (N)correction. The corrected pressure is obtained by adding this correction to
the pressure from simulation at some N. Inset: Effect of temperature on the finite-size effects of the corrected
pressure. Lines are linear fits weighted by uncertainties.
Figure 5.9 demonstrates the size dependence of the pressure. The main figure shows both the
pressure as measured in simulation (direct) and the corrected pressure using the harmonic HarmFSC
correction (proposed initially here [32] to measure FSC of FE) which is defined as Pharm(∞)−
Pharm(N)with Pharm (∞)obtained by extrapolation versus 1/N. Thus, the finite-size dependence of
the corrected pressure is only due to the anharmonic pressure (lattice pressure is independent on N).
It can be noticed that the slope of the corrected pressure is remarkably smaller (almost flat) than the
direct one; hence, a quick (but accurate) estimate of the pressure in the 1/N →0limit can be obtained
using only single simulation of a relatively small system size. In contrast, several points are needed
for the direct method in order to extrapolate to that limit. Similar behavior was observed previously
[32, 33, 34] in measuring the FSC of the FE. The effect of temperature on the size dependence of the
corrected pressure (off the infinite limit for each T) is depicted in the inset of Fig. 5.9. A pattern of
large slope at high temperatures can be noticed; thus the finite size effects of the corrected pressure is
more important as the system deviates from harmonicity. However, the slope of the corrected pressure
at the highest temperature visited (T= 0.8) is still much smaller (≈6times smaller) than that of the
75
0 0.01 0.02 0.03 0.04 0.05 0.06
rc
-3 (LJ units)
-2.818
-2.817
-2.816
-2.815
-2.814
-2.813
-2.812
-2.811
Pressure (LJ units)
LRC
LatLRC
LatLRC + HarmLRC
ρ = 1.0 , T = 0.2
rc = 2.5
rc = 7.5
Figure 5.10: (Color online) Long-range effects of measuring the pressure using three corrections: the standard
LRC (assuming homogeneous medium), lattice LatLRC ≡Plat(∞)−Plat (rc), and harmonic HarmLRC ≡
Pharm(∞)−Pharm (rc). The corrected pressure is obtained by adding the correction to the pressure from
simulation at some rc. Note that some points from LRC and LatLRC with short rcare out of the scale.
direct one (see the main figure).
In order to estimate the thermodynamic limit of the pressure extrapolation to infinite cutoff is
needed. The effect of truncation on the pressure is depicted in Fig. 5.10 using different LRC ap-
proximations. Direct measure of the pressure is not presented as it goes beyond the figure region.
As a first approximation, we added the standard LRC [1] (which assumes homogeneous medium) to
the measured pressure. However, this approximations shows a large deviation from the infinite rc
limit (rc= 2.5and 3.0results are out of the scale). Second, we used a lattice correction (LatLRC),
which is defined as Plat(∞)−Plat (rc). A truncation of rc= 300 is used to estimate the infinite
lattice pressure as it showed well-convergence result (to the eighth digit). However, its slope is still
noticeable (with the rc= 2.5out of the scale). To improve this correction even more, a harmonic
correction (HarmLRC), defined as Pharm(∞)−Pharm(rc), is added. Thus, the cutoff dependence of
this “lattice+harmonic” correction is only due to the anharmonic contribution. The figure shows that
the slope is remarkably smaller (statistically flat); thus, using small cutoff (as small as rc= 2.5) gives
a quick (but accurate) measure of the 1/rc→0limit. In contrast, cutoffs of at least rc= 6.5is
76
0510 15 20 25
Thousands of MC steps
0.0
0.2
0.4
0.6
0.8
1.0
Energy autocorrelation
Conventional
Harmonically mapped
ρ = 1.0 , Τ = 0.2
Figure 5.11: (Color online) Normalized autocorrelation function of the conventional and mapped measurements
of the energy. A trajectory of 2×105samples is used.
required for other corrections to get statistically similar results, but with about 17 times more effort
than using rc= 2.5. Not like the finite-size effects, the temperature has a negligible role on the anhar-
monic pressure versus rcrelation; running MC at different temperatures (including melting) shows
statistically flat relation.
5.3.5 Effect of Mapping on the System Trajectory
Although Fig. 5.1 shows small standard-deviation of the harmonically-mapped energy relative to
the conventional one, the uncertainty of the average depends on the correlation in the data as well.
Figure 5.11 shows, qualitatively, less correlation of the harmonically-mapped MC trajectory than the
conventional. Quantitatively, however, the number of “independent” samples (see figure caption for
definition) versus temperature is depicted in Fig. 5.12. Enhancement (by a factor of about 2−4) of the
number of independent samples, using harmonically-mapped MC trajectory, is observed. In addition
to the effect of that on decreasing the uncertainty, a meaningful statistics can be obtained with 2−4
times fewer samples. A similar conclusion (however with less improvement) is observed using MD
simulation (not shown).
77
0 0.2 0.4 0.6 0.8 1
T/Tmelt
0
5
10
15
20
25
Independent samples (%)
Conventional
Harmonically-mapped
ρ = 1.0
Figure 5.12: (Color online) Number of independent energy samples out of 2×105measured samples. The
number of independent samples is estimated as Var/σ2with Var and σthe sample variance and uncertainty,
respectively.
In addition, the equilibaration rate of both averaging methods is depicted in Fig. 5.13. The in-
stantaneous energy difference (from the average) is normalized by the initial energy difference of
each method in order to distinguish between the equilibaration speed and accuracy. We used the time
needed for that normalized energy difference to be equal to the uncertainty from a production run
as our “equilibaration” time. Using normalized uncertainties from MC production run of 104trials
(dotted lines) we get about 3times smaller equilibaration time with the mapped average.
Finally, we studied the effect of mapping on the energy convergence rate with MD time-step size
(∆t). Figure 5.14 shows a substantial improvement of the convergence rate using the mapped average;
therefore, larger time-step can be used without loosing energy accuracy. This is especially important
with models like DFT where the cost of each MD step is expensive; thus, using such mapped average
can visit much longer dynamics than the conventional method.
78
0 20 40 60 80
Thousands of MC steps
0.001
0.01
0.1
1
(U-Uavg)/(U0-Uavg)
Conventional
Harmonically mapped
normalized uncertainty (mapped)
normalized uncertainty (conventional)
ρ = 1.0 , Τ = 0.2
Figure 5.13: (Color online) Effect of mapping on the equilibaration rate of conventional and harmonically-
mapped averagings. All data are normalized to the initial energy U0(off the average Uavg). The dotted lines
are the normalized uncertainties from a 104MC production run.
0 0.002 0.004 0.006 0.008 0.01
Integration step size, ∆t (LJ units)
-7.8425
-7.8420
-7.8415
-7.8410
-7.8405
Molar energy (LJ units)
Conventional
Harmonically mapped
ρ = 1.0 , T = 0.2
Figure 5.14: (Color online) Convergence of the average energy with respect to MD time-step size. Red line
simply joins the points. All points are generated using the same integration time, t= 5000 (LJ units).
79
5.4 Application to embedded-atom model
(EAM)
Here, we test the generality/applicability of the mapped-average method to many-body models. An
embedded-atom model of iron [146] is used as an example. The model parameters are generated from
fitting first-principle (linear muffin tin orbital method) calculations data. The EAM model configura-
tional energy is given by
E=X
i
Ei(5.17)
where Eiis given as a sum of pairwise (φ(rij )) and many-body (embedding, F(ρi)) contributions as
following
Ei=1
2X
j;j6=i
φ(rij) + F(ρi)(5.18)
with the embedding charge given by this sum
ρi=X
j;j6=i
ρ(rij)(5.19)
The functions φ,ρ(rij ), and Fare given by
φ(rij) = ǫ(a/rij )n(5.20)
ρ(rij ) = (a/rij )m(5.21)
F(ρi) = −ǫC X
j
√ρj(5.22)
Where n,m,ǫ,a, and Care adjustable parameters; their specific values are given here [146]. A
106-steps MD simulation simulation is used to generate sample the configuration space with a cutoff
distance of 6.0˚
Ais used. The method is first tested against the stable hcp crystal structure of iron.
Figures 5.15 and 5.16 shows the improvement HMA introduces (for both Uand P) along isochore
80
0 1000 2000 3000 4000 5000 6000
Temperature (K)
10
Error ratio (conventional/mapped)
U
P
hcp , N = 250
v = 7.0 Å 3/atom
PLat = 242 GPa
Figure 5.15: Pressure and energy uncertainty ratios of EAM model along v = 7 ˚
A3/atom isochore.
and isotherm, respectively, up to states close to melting. The behavior is, qualitatively, similar to
that of LJ model. The efficiency of HMA method (both Uand Pbehaves similarly) increases with
temperature and density. The CPU time saving range (depending on state) is ≈10−1000 times along
the isochore and ≈10 −50 times along the isotherm.
Moreover, the method is tested against bcc structure of iron which is known to be unstable in
the high pressure region (at least for low temperatures). Figure 5.17 shows measurement of anhar-
monic contribution of the configurational energy along an isochore of PLat = 107 GPa (where bcc is
known to be unstable). It is interesting to notice that although there is structural change of the bcc at
temperatures lower that ≈500 K, the HMA still provides and accurate estimation of the energy.
Finally, we check the effect of MD integration time-step (dt) on the convergence of energy as
depicted in Fig. 5.18. Although using a step of 1fs gives large scattering the the data, the HMA energy
is much less scattered using same time step. In addition, decreasing dt to only 0.25 fs improves the
conventional at high temperature while the data still largely scattered in the very low-Trange. The
HMA data from both time-steps are indistinguishable from each other. Thus, with HMA a longer
dynamics can be visited using relatively large dt while keeping accurate results.
81
6.6 6.8 77.2 7.4 7.6 7.8 8
v (Å 3/atom)
3
4
5
6
7
8
Error ratio (conventional/mapped)
U
P
hcp , N = 686
T = 3000 K
Figure 5.16: Pressure and energy uncertainty ratios of EAM model along T= 3000 K isotherm.
5.5 Conclusions
In conclusion, we showed that knowledge of the approximate (i.e. harmonic) dependence of molec-
ular structure on state can be used to formulate alternative ensemble averages, separating known
behavior from averages that quantify deviations from it. Then, direct measurement of the deviations
(i.e. anharmonic) by molecular simulation is done absent noise produced by behavior that is already
well characterized. The scheme is applied to crystalline solids, where lattice dynamics provides the
approximate description. The approach yields thermodynamic properties of a given precision with a
hundredfold or more reduction in computational effort when applied to LJ and EAM models
The methods developed here can be extended to handle crystalline systems in other contexts:
•Extension to molecular crystals should be possible, and effective to the extent that the crystal
still has harmonic character. The appropriate mapping for rotation is not obvious, but there is
work to build on when attempting this [34].
•The treatment presented here has been entirely classical. Nuclear quantum effects can be han-
dled using semiclassical or path-integral methods [147]. It is likely that harmonically-mapped
averaging can be used to some benefit when applied to such simulations. However, the temper-
82
0500 1000 1500 2000 2500 3000
Temperature (K)
-14
-12
-10
-8
-6
-4
-2
0
Uanh/N (meV)
conventional
HMA
bcc , N = 1024
v = 8.33 Å 3/atom
PLat = 107 GPa
Figure 5.17: Conventional and HMA estimations of the specific anharmonic energy of BCC structure of iron
along v = 8.33 ˚
A3/atom isochore.
ature dependence exhibited by quantum harmonic systems differs markedly from the classical
form, so the mapping given by Eq. (5.11) will require reformulation.
•Harmonically mapped averages of the energy and heat capacity can be performed in N P T
simulations, in the same manner as described here for N V T . There is of course no need to
compute the pressure then, but the underlying concept can be applied to enhance sampling of
the volume [143].
•In terms of properties, we have focused here on an important but not comprehensive set of solid-
phase behaviors, and the method could be extended to handle others. In particular, anisotropic
mappings can be formulated to improve calculation of other elastic constants. A noteworthy
feature of the method is that all mapped averages can be measured simultaneously, because the
enhancement is not targeted at modifying sampling. Another property of importance is the free
energy, which is needed to gauge thermodynamic stability of the crystals.
83
0 1000 2000 3000 4000 5000 6000 7000
Temperature (K)
-10
-8
-6
-4
-2
0
Uanh/NT2 (10-10 eV/K2)
conventional: dt=1.0 fs
HMA: dt=1.0 fs
conventional: dt=0.25 fs
HMA: dt=0.25 fs
HMA fitting: dt=1.0 fs
HMA fitting: dt=0.25 fs
hcp , N = 250
PLat = 242 GPa
v = 7.0 Å 3/atom
Figure 5.18: Effect of MD time step (dt) on the accuracy of measuring EAM anharmonic energy using conven-
tional and HMA methods.
84
CHAPTER 6
Thermodynamic Stability of Iron Polymorphs in the Earth’s Inner-Core
6.1 Introduction
6.1.1 The Schr¨
odinger Equation
For atomic or molecular system, the electrons are fully descried by Schr ¨odinger’s wave equation.
Due to the large mass difference between the nuclei and electron, the electrons interactions can be
separated from those of atoms; this is called Born-Oppenheimer approximation. The fundamental
time-independent Schr¨odinger equation of an isolated system of Nelectrons is given by [148]
ˆ
HΨ = EΨ(6.1)
where Ψ = Ψ({r};{R})is the wave function that depends explicitly on all electronic positions {r}
and para-metrically on all ionic positions {R},Eis the electronic energy, and ˆ
His the system’s
Hamiltonian operator of a system of N-electrons, each of mass meand change e,
ˆ
H=−~2
2meX
i∇2
i−X
i,I
ZIe2
|ri−RI|2+1
2X
i6=j
e2
|ri−rj|(6.2)
where iand Iare subscripts of electrons and nuclei, ZIis the atomic number of nucleus I. The first
term represents the electronic kinetic energy, second term is the electron-nucleus attraction energy,
last term is the electron-electron repulsion energy. In this approximation, the total energy can be
simply obtained by adding the electronic to ionic contributions. Unfortunately, Schr¨odinger equation
only solvable, analytically, for very simple systems (e.g. hydrogen atom). Numerical approaches
(e.g. finite difference), on the other hand, can not handle such multidimensional partial differential
equations.
One of the early approaches to tackle this problem was using variational principle to get the ground
state energy. This was applied in 1928 by Hartree [149] using independent-electron approximation
and then by Fock [150] in 1930 using the Slater determinant (the method is known as Hartree-Fock
method). However, those are mean-field approximations relying on the independent-electron hypoth-
esis; which is not true for highly correlated quantum systems. Therefore, a completely different
“paradigm” was needed.
6.1.2 Density-Functional Theory (DFT)
A remarkable step towards solving this problem was accomplished using the density-functional the-
ory (DFT). In this theory, the problem of getting the complicated N-electron wave function Ψfrom
Schr¨odinger equation is replaced by a much simpler problem problem that involves only the electron
density ρ(r)that is function of only one argument. The original idea is dated back to the works of
Thomas [151] and Fermi [152] who applied variational principle, using the electron density as the
trial function, assuming non-interacting electrons (i.e. uniform electron gas). It was not before 1960s
that the idea of existing a one-to-one mapping between Ψand ρwas rigorously proven in a form of
two theories by Hohenberg and Khon [153].
In 1965, a remarkable breakthrough was achieved by Kohn and Sham [154] who invented an ap-
proach to implement the Hohenberg-Khon idea in practice. This became known later by Khon-Sham
DFT method. In this approach, the actual (interacting) system is replaced by a fictitious system of
non-interacting particles that is responsible for most of the physics leaving a small residual correc-
86
tion (called exchange and correlation, “XC”) that can be treated based on the system. The theory is
in principle exact (given an exact XC functional) Applying the variational principle on such system
yields the following a simpler None-electron independent differential equations
−~2
2me∇2
i+veff (r)ψi=ǫiψi(6.3)
where veff is the effective potential in which the non-interacting electrons move, it is given by
veff =v(r) + Zρ(r′)
|r−r′|dr′+vxc(r)(6.4)
with v(r)the external (electron-ions interactions) potential, second term is the Hartree potential, and
vxc(r)is the exchange-correlation potential. The electron density can be computed using
ρ(r) = X
i|ψi(ri)|2(6.5)
Since veff depends on ρ(r) through Eq. 6.4, Eqs. 6.3-6.5 must be solved self-consistently. Figure 6.1
shows a schematic flowchart of how KS equations can be solved.
6.1.3 Solving KS equations in periodic systems
The periodicity of crystalline systems allows using plane waves to solve the KS differential equations.
This ability of doing this is based on Bloch theorem of any periodic Hamiltonian (e.q. 6.3), which
states that the solution of such Hamiltonian must satisfy this form
ψk(r) = eik·ruk(r)(6.6)
where kis any allowed wave vector (usually chosen to be in the first Brillouin zone) for the specific
crystal structure, and uk(ris arbitrary functions with the periodicity of the lattice. Since uk(ris
periodic, taking its discrete Fourier transforms gives
ψk(r) = X
G
ck+Gei(k+G)·r(6.7)
where Gis the reciprocal-lattice vector and the constants ck+Gare essentially the solution of KS
equation in the reciprocal space.
87
Figure 6.1: Flowchart of solving KS equation iteratively.
In principle, infinite number of plane waves are needed; however, contribution from higher Fourier
components (i.e. large |k+G|) is small. Thus, in practice, the expansion is truncated at some kinetic
energy cutoff Ecut such that
~2|k+G|
2me≤Ecut (6.8)
Many quantities (e.g. energy and pressure) involve integration over k; thus, the accuracy of DFT
depends on k-mesh density over which the KS equations are solved. However, for metallic solids (as
our case, i.e. Fe) a sharp discontinuity exists in the k-space (i.e. Fermi surface is discontinuous).
Hence, many k-points is needed to avoid self-consistence convergence problems.
6.1.4 Finite-temperature DFT: Mermin’s approach
The above DFT analysis is valid to get ground-state properties only in the T= 0 K limits. How-
ever, Mermin [155] extended KS equations to handle finite-temperature problems by including the
88
electrons occupancy through Fermi-Dirac occupation function,
f(ǫi) = 1
1 + eǫi−ǫF
σ
,(6.9)
where σ≡kBTand ǫFis Fermi energy. In this case, the functional that need to be minimized is no
longer the energy (as the case with T= 0 KS equations), but rather the free energy Fof the system
at temperature T. The free energy is given by F=E−T S , where Sis the electronic “entropy” and
given by
S=−kBX
i
[filn fi+ (1 −fi) ln (1 −fi)] (6.10)
The Mermin finite-temperature approach can serve to measure a physical energies at finite temper-
ature or as a smearing method to get well converged results using relatively small number of k-points.
For our purposes, we use it as a mean to get physical quantities.
6.2 Reweighting
Ab initio molecular dynamics simulation is computationally rather expensive. In addition, metallic
systems require order of magnitude denser k-grid in order to handle the sharp discontinuity of Fermi
surface. Therefore, sampling the actual DFT potential, directly, in MD simulation is not very practical
approach. However, using reweighting techniques a “cheap” (less accurate DFT) model can be used
to sample the system and measure any “expensive” thermodynamic quantity Ovia the following
reweighting formula
hOie=Oe−β∆Uc
he−β∆Uic
and ∆U≡Ue−Uc(6.11)
where the subscripts “e” and “c” represent the expensive and cheap potentials, respectively. Similarly,
the free energy difference between the two models can be computed using
∆A≡Ae−Ac=−kBTln e−β∆Uc(6.12)
Equations 6.11 and 6.12 are similar to the free-energy perturbation formula (Eq. 2.7); thus, the
technique is only useful if the cheap and expensive models are “close” to each other; i.e. have large
89
overlap. The optimum case is if both potentials are different, systematically, by a constant; in this
case hOie=hOic; i.e. all samples contribute equally to the average. However, in practice, both
models are different by more than a constant; in this case, different samples contribute to the average
with different weight (proportional to e−β∆U). Thus, the number of “effective” sample we get from
the cheap trajectory will be always less than the actual samples we collect.
6.3 Crystal Structure of Earth’s Inner Core
(EIC)
Knowing the crystal structure of the Earth’s inner-core (IC) is essential in understanding how the
planet was evolved. Although it is broadly accepted that the IC is in a solid state and composed of
mainly iron (Fe)[156, 157], its crystal structure is still uncertain to large extent, both theoretically
and experimentally. Due to the extreme pressure and temperature (330 −360 GPa and 5000 −7000
K[158]) condition of the IC, it has been a challenge for the experimentalists to bring those conditions,
simultaneously, in the lab. Theoretically, on the other hand, at such elevated temperature the entropic
free-energy is presumed to contribute, non-negligibly, to the stability of the competing structures.
Moreover, the magnetization nature of iron and the possibility of existence of light elements (e.g.
Ni[156], C and S[159]) in the IC add an extra complexity to the problem.
Iron is well-known to exist in four different polymorphs at low pressures. At ambient condition
it stabilizes, by magnetization, in a ferromagnetic body-centered cubic (bcc) α-Fe up to 1185 K
where it turns into a paramagnetic (TC= 1064 K) face-centered cubic (fcc) γ-Fe and eventually
to a paramagnetic bcc δ-Fe at 1667 K before it melts at 1811 K. At low temperature, as the α-Fe
is compressed it transforms into a paramagnetic hexagonal-close packed (hcp) ǫ-Fe at 13 GPa and
persists to extremely high pressures at room temperature (experimentally up to 304 GPa[160] and
theoretically up to 5000 GPa[161]). However, entropic effects at high temperatures might change this
90
picture, as the case of other transition metals (e.g. Ti, Zr, and Be[162]), which transforms, just below
melting, from close-backed structure to more open structure, bcc.
Although it is not possible to test the crystal structure of the IC directly, earthquake experiments
provide useful information that can put some constraints on the IC candidate structures. Seismolog-
ical data[163] suggested that the IC is elastically anisotropic −the seismic waves propagate faster
(3−4%) along the Earth’s spin axis than the perpendicular direction. Thus, due to the preferred
orientation of the hcp structure, Stixrude et al.[164] used first-principles calculations to show that
single-crystal hcp can reproduce elastic anisotropy similar to that of the IC. However, they used a
fixed c/a ratio which was shown later not to be the case[165, 166, 167, 168, 169]. However, the
existence of hcp-phase at the IC condition was first questioned in 80’s by shock-wave experiments
of Brown and McQueen[170]. They observed a sharp discontinuity in the shock-wave velocity at
200 GPa and 4400 ±300 K which was attributed to a solid-solid transformation (ǫ-γas they ini-
tially expected). Shortly, using diamond-anvil cell (DAC) experiment, at lower pressures and tem-
peratures, Boehler[171] was able to extrapolate the results to the Brown and McQueen’s state and
conclude the possibility of having solid-solid transition, however to a new phase (other than γ) of un-
known structure. The solid-solid transition was sooner confirmed, around 160 GPa, for temperatures
above 2000 K.[172] Interestingly, bcc was later predicted, to the first time, as the crystal structure
of that new phase (and the IC) using quasi-harmonic approximation of a simple classical pairwise
model.[173] Same conclusion was later made using inference from shock waves data at P > 200
GPa and T > 4000 K.[174] However, post-hcp transition was not detected by later shock-wave
experiments.[175, 176] Besides, very recently, advances in laser-heated DAC techniques made it pos-
sible to bring the IC state in the lab.[177] In those experiments, the hcp structure was the only structure
observed for pure iron to states of: 200/5000[178], 273/4490[166], 301/2000[179], 340/4700[167]
, and 377/5700[165] (GPa/K). A conclusion which was confirmed by recent multiple-shocks experi-
ment (up to 560 GPa and 8000 K).[180]
First-principle calculations, on the other hand, provide an efficient alternative that can help resolv-
ing the controversy among experiments. At 0K, density-functional theory (DFT) calculations is well-
91
known to favor hcp structure at the IC pressure for several reasons: hcp has lower energy/enthalpy[181,
182, 183, 184, 185, 161] , it is stable, both mechanically[184, 185] and vibrationally [186, 187](i.e. no
imaginary frequencies) and finally, as mentioned above, due to its preferred orientation that can mimic
Earth’s core anisotropy.[164] The other candidate, bcc, was ruled out due to its mechanical[181, 183,
184] vibrational[188, 189, 185, 187] instabilities at high pressures (above ∼150 GPa, at 0K). How-
ever, DFT calculations showed that only bcc retains finite magnetic moment at the IC pressures[184,
190, 191, 192], thus, the magnetic entropy[182] can favor its stability at the IC temperatures. More-
over, at such high temperatures the entropy (electronic and vibrational) may play a rule in stabilizing
bcc polymorph (as the case of Ti, Zr, and Be[162]). Thus, extensive finite-temperature ab initio
molecular dynamics (AIMD) simulations were then followed to study the IC state.
At finite temperatures, the mechanical stability of bcc was first tested by Vo˘cadlo[189] using
AIMD. Based on hydrostatic equilibrium condition (isotropic tetragonal stresses), they concluded
that bcc becomes mechanically stable for T > 3000 K at P∼300 GPa, a result which was
then verified using same argument.[193, 194, 195] However, this criterion does not necessarily cor-
respond to a minimum free energy[185]. Explicit measurement of the absolute Helmholtz free-
energy versus tetragonal strain (at ∼260 GPa) showed that bcc becomes mechanically stable at
6000 K.[185] However, last result was questioned due to its lack of accuracy and convergence[196]
and low precision[197]. Using thermodynamic integration of stresses, along the Bain path, to get the
relative Helmholtz free-energy between bcc and fcc, it was shown that bcc is mechanical unstable at
similar conditions.[196] Same conclusion was drawn through direct measurement of the tetragonal
shear modulus, which shown to be negative for T= 5500 −7000 K at IC pressures;[196] contrary
to a previous estimate[198] of the shear modulus (using smaller box). On the other hand, vibrational
stability of bcc at finite-temperature was recently[199] tested using a method called self-consistent ab
initio lattice dynamic (SCAILD)[200]. In the context of this method bcc showed vibrational stability
just below the melting line at pressures relevant to the IC. In addition, it is interesting to mention here
that using EAM model fitted to well-tested DFT calculations, bcc structure showed anisotropic behav-
ior, despite its high symmetry, at IC conditions which can explain the Earth’s elastic anisotropy.[201]
92
Moreover, single-crystal hcp system failed to reproduce such anisotropic behavior at IC state;[202]
contrary to other calculations (but at 0K).[164]
To attain full stability, free energies of the IC candidates must be obtained to assess thermodynamic
stability. As a first approximation, Vo˘cadlo et al.[203, 186] used quasi-harmonic analysis (with DFT)
to show that hcp (bcc was not included due to its imaginary modes) is the most stable phase at IC
conditions. However, electronic excitation contribution was not included in their force constants;
including that makes fcc more stable at 5000 −6000 K and 300 −360 GPa, although the free-energy
differences are very small.[204] On the other hand, hcp-bcc transition was first predicted, using EAM
model fitted to DFT data (assuming zero entropy difference between bcc and hcp), for states just below
melting curve at high pressures.[205] Recently[158], Belonoshko et al. showed that both bcc and hcp
structures have identical free-energies, within error bars, using self-consistent phonon calculations.
However, based on full free-energy DFT calculations (but using approximate formula of the exact
thermodynamic integration), hcp is marginally stable over bcc[189] and fcc[206], with very small
free energy differences (35 and 14 meV from bcc and fcc, respectively). Besides, the presence of
small light elements was expected to stabilize bcc, experimentally[207] and theoretically.[189]
6.4 Simulation Details
Ab initio molecular dynamics (AIMD) simulations were performed using projector augmented-wave
(PAW) [208] pseudopotential, as implemented in VASP [209], within the generalized gradient ap-
proximation (GGA) using the PBE parametrization [210]. Canonical NVT ensemble is used with the
temperature controlled using Andersen thermostat [211]. Only hexagonal close packed (hcp) is used
in in this study. Supercells composed of 3×3×3primitive hcp unit cells (N= 54 atoms) are used.
An energy cutoff of 250 eV was used with 3p, 3d, and 4sorbitals treated as Fe valence states.
Integration over the first Brillouin-zone (BZ) was done using 2×2×2Monkhorst-Pack k-point grid.
First-order Methfessel-Paxton electronic smearing, with the smearing width of σ= 0.2eV, was used
to improve DFT convergence. These parameters are not converged within the meV range because
93
0 2000 4000 6000 8000
Temperature (K)
1
10
Error ratio (conventional/mapped)
U
P
hcp , N = 54
v = 6.35 Å 3/atom
PLat = 338 GPa
Figure 6.2: Pressure and energy uncertainty ratios of DFT model along v = 6.35 ˚
A3/atom isochore of DFT
model.
the purpose here is to show the efficiency of the HMA method. However, if converged results are
of interest, snaps can be taken from the MD trajectory and more converged DFT calculations can be
computed for them; then, the free energy can be corrected using the reweighting technique (Eq. 6.12).
6.5 Results and Discussion
A thermodynamic integration along temperature and volume paths is required to reach to the IC
state (≈360 GPa and ≈6000 K[158]). For the purpose of this thesis, we run MD simulations at
temperatures from 1000 to 7000 K along an isochore of v = 3.5˚
A3(PLat = 338 GPa). Then, another
set of MD simulations were run along an isotherm of T= 6000 for densities equivalent to pressure
range of ≈100 −400 GPa. Using these two paths (along with the harmonic free energy) we were
able to get the absolute free energy along that isotherm.
A precise estimate of pressure and anharmonic energy is, then, needed to get a FE estimate that is
within the meV error range. Thus, we will show the effect of using HMA on the uncertainty in these
thermodynamic quantities. Figure 6.2 represents the uncertainty reduction due to using HMA method
94
6.4 6.6 6.8 77.2
v (Å 3/atom)
1
1.5
2
2.5
3
3.5
4
4.5
Error ratio (conventional/mapped)
U
P
hcp , N = 54
T = 6000 K
Figure 6.3: Pressure and energy uncertainty ratios of DFT model along T= 6000 K isotherm.
along the isochore. Like the LJ and EAM, the difficulty of measuring energy and pressure decreases
at low temperatures and high densities. Similarly, Figure 6.3 shows the improvement along the T=
6000 K isotherm. Although that temperature is considered high (relative to melting point), the HMA
still improves the energy and pressure estimations (at least 4 times cheaper that the conventional). It is
also observed that the pressure estimate is even better (i.e. less noisy) than the energy estimate along
that isotherm.
Next, we show the actual variation of TI integrand (i.e. anharmonic energy over T2and pressure)
using both HMA and conventional methods. Figure 6.4 represents the anharmonic configurational
energy along an isotherm of relevant density to the IC state. The HMA data looks more precise
and consistent with each other while the direct measurement of energy is noisier and systematically
inconsistent with itself and with the HMA data. The behavior is due to using dt = 1 fs; testing the
energy convergence with different timesteps shows that 1 fs is relatively large — a behavior that was
previously observed with EAM model (see Figure 5.18). The consistency of the HMA results allowed
us to fit it using second-order polynomial (red line), with fitting parameters given in the figure caption.
The off-lattice pressure variation versus volume is depicted in Figure 6.5 for both conventional
95
0 1000 2000 3000 4000 5000 6000 7000
Temperature (K)
1.5×10-9
2.0×10-9
2.5×10-9
3.0×10-9
3.5×10-9
Uanh/NT2 (eV/K2)
conventional
HMA
HMA: fitting
hcp , N = 54
v = 6.35 Å 3/atom
PLat = 338 GPa
Figure 6.4: Anharmonic energy along v = 6.35 ˚
A3/atom isochore of DFT model. Red line is a second-
order polynomial fit of the HMA data. The fitting parameters (in increasing polynomial powers) are a0=
1.35(7) ×10−9eV/K2,a1= 3(5) ×10−14 eV/K3, and c2= 2.6(6) ×10−17 eV/K4.
6.4 6.6 6.8 77.2
v (Å 3/atom)
33
34
35
36
37
38
39
P - PLat (GPa)
conventional
HMA
HMA: fitting
hcp , N = 54
T = 6000 K
Figure 6.5: Conventional and HMA measurements of pressure along T= 6000 K isotherm of DFT model.
The line is a second order polynomial fit of the HMA data with coefficients a0= 217.6(3) GPa, a1=−59(8)
GPa/ ˚
A3, and a2= 4.8(6) GPa/ ˚
A6.
96
6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0
v (Å 3/atom)
100
150
200
250
300
350
400
450
PLat (GPa)
data
fitting
6.5 77.5 8
-1
-0.5
0
0.5
1
Fitting residual (GPa)
hcp , N = 54
Figure 6.6: Lattice pressure versus volume of DFT model. The red line is fitting using third-order Birch-
Murnaghan equation of state. The fitting constants (see Eq. 6.13) are c0=−272.53 GPa, c1= 39969.30
GPa/ ˚
A5/3,c2=−239426.00 GPa/ ˚
A7/3, and c3= 520974.00 GPa/ ˚
A3. Inset: the error in pressure due to
fitting; x-axis is the same as the main figure.
and HMA approaches. The pressure estimate from HMA method is, again, more precise along the
entire densities studied here. On the other hand, the conventional data look systematically higher than
the HMA results; which is due to timestep size. The HMA off-lattice pressure is then fitted using a
second order polynomial with fitting parameters given in the figure caption.
The full pressure (lattice and off-lattice components) is needed in order to get free energy differ-
ences along the T= 6000 K isotherm. Figure 6.6 shows the lattice pressure using single-point DFT
calculations (with expensive parameters). The data were, then, fitted to Birch-Murnaghan equation of
state of third-order in the form
PLat =c0+c1V−5
3+c2V−7
3+c3V−3(6.13)
where ciare the fitting constants and given in the figure caption.
The absolute free energy estimation along the T= 6000 K isotherm is shown in Figure 6.7. Pres-
sures and energies used to generate the plot are measured using the HMA approach; comparison of FE
using HMA and conventional methods is not reported here as we showed already that comparison for
97
350 375 400 425 450 475
P (GPa)
-9
-8
-7
-6
-5
A (eV/atom)
350 375 400 425 450 475
-3.40
-3.35
-3.30
-3.25
A - ULat (eV/atom)
qh + anh
qh
lattice
qh
qh + anh
hcp , N = 54
T = 6000 K
Figure 6.7: Helmholtz free-energy of DFT model including from lattice, quasi-harmonic (qh), and anharmonic
(anh) contributions. The anharmonic data are obtained from HMA method. Inset:
the FE integrands (i.e. energy and pressure). The quasiharmonic (qh) free energy results are shown
for comparison. The inset shows the full and qh free energies relative to the lattice energy. It can be
noticed that the error in measuring the anharmonic FE is small compared to the difference between
the qh and full FE.
6.6 Conclusions
In conclusion, we applied the mapped averaging technique to estimate the free energy of iron at high
pressures and temperature regime relevant to the Earth’s inner-core states. The Earth’s inner core is
known to be composed of pure iron. Ab initio molecular dynamic simulations were used to sample the
system. Just like the LJ and EAM models, the HMA method showed substantial improvement (over
conventional method) of measuring thermodynamic quantities (energy, pressure, and free energy)
even at states close to iron melting. The free energies along T= 6000 K were computed along the
lattice and quasiharmonic contributions. The error of estimating the anharmonic contribution of FE
is within the meV range. Thus, the HMA can be effectively used to be able to resolve the free-energy
98
differences between different iron candidates in the inner-core.
99
CHAPTER 7
General Conclusions and Future Work
In this dissertation, we have developed highly efficient molecular simulation methods to compute
thermodynamic properties of crystalline system. The methods are applied to a wide range of in-
termolecular interaction models, viz. hard spheres (Chapter 3), TIP4P model of clathrate hydrates
(Chapter 4), Lennard-Jones and embedded-atom models (Chapter 5), and density-functional theory
model of iron (Chapter 6). Although the specific conclusions are given for each chapter, we will
present here the most original findings of this work.
In Chapter 3, we have introduced a comparative study of measuring the absolute Helmholtz free
energy of FCC structure of hard-spheres model. We consider all combinations of three choices defin-
ing the methodology: (1) the reference system: Einstein crystal (EC), interacting harmonic (IH), or
r-12 soft spheres (SS); (2) the integration path: Frenkel-Ladd (FL) or penetrable ramp (PR); and (3)
the free-energy method: overlap-sampling free-energy perturbation (OS) or thermodynamic integra-
tion (TI). The choice between TI and OS is a matter of convenience as both give very similar behavior;
with advantage of TI method due to its simplicity. Among the other 6routes, we find that using SS as
a reference along with PR path provides the best choice to compute the free energy of hard spheres.
This choice reduces the computational effort by about 6times less than the widely used EC+FL+TI
combination. In addition, Both the SS and IH references show great advantage in capturing finite-size
effects, providing a variation in free-energy difference with system size that is about 10 times less
than the well-known EC reference. This allowed us to compute the free energy in the 1/N →0
limit using relatively small system. Although the method is applied to hard spheres, it can be easily
extended to more complex hard potentials; e.g. densely packed tetrahedra.
In Chapter 4, efficient lattice dynamics techniques are utilized to compute the harmonic free en-
ergy (quantum and classical) of clathrate hydrate phases, viz. the cubic structure I (sI), cubic structure
II (sII), and hexagonal structure H (sH) phases, in the absence of guest molecules. TIP4P model is
used to describe water molecules interactions. We examined in particular the effects of finite size and
proton disorder on the calculated free energies. We fond that finite-size effects due to proton disorder
of hydrates is small relative to that of any particular H-isomer configuration We find that at 300 K,
the finite-size, proton-disorder, and quantum effects between phases are, respectively, on the order
of 0.4, 0.03, and 0.1 kJ/mol. The harmonic free energy can be used as a reference to compute the
full (harmonic and anharmonic) free energy at high temperatures; e.g. using methods described in
Chapter 5.
In Chapter 5, we introduced a very general and rigorous theory to compute thermodynamic prop-
erties of, essentially, any phase of matter. The method relies on the existence of an analytical (or
numerical) solution of the thermodynamic property of interest and then use molecular simulation
to compute the small contribution off the known solution. This is achieved by mapping the coor-
dinates according to the behavior of the known solution. Here, we apply the mapping method to
crystalline systems, utilizing the harmonic nature of solids; thus the method developed here is called
harmonically-mapped averaging (HMA). The method is used to get first- (i.e. energy and pressure)
and second- (i.e. heat capacity and bulk modulus) order derivative of the free energy with respect to
temperature and volume, respectively. Calculation of the FE can benefit from mapped averaging when
used in conjunction with thermodynamic integration. The low-temperature harmonic system provides
a convenient reference for computing the FE via integration in temperature. Such a scenario exploits
the tremendous efficiency of mapped averaging at low temperature, even if the FE being calculated is
101
close to melting. The HMA is applied to both pairwise potential (LJ model) and to a many-body po-
tential of iron (EAM model) with qualitatively similar behaviors in computational efficiencies using
HMA method.
Finally, in Chapter 6, the HMA method is applied to a more rigorous and realistic problem, viz.
free energy of iron in the Earth’s inner-core. Using thermodynamic integration along an isotherm at
temperature relevant to the Earth’s core state (6000 K), were were able to estimate the absolute free
energy of hcp candidate withing meV error bars. The free energy of other states and iron candidates
(e.g. bcc and fcc) can be easily done to be able to predict the crystal structure in the Earth’s inner
core.
Considered more broadly, the notion of mapped averaging as a means to evaluate thermodynamic
properties is a very general one, and could find application to systems other than crystalline solids.
Indeed, the approach presents a relatively unexplored route to formulate improvements in simulation
technique. Creativity and physical insight are needed to construct a good mapping for a given sys-
tem, but idealized models exist for many behaviors, and these can be examined for inspiration when
attempting to extend the approach.
102
APPENDIX A
Atomic-to-molecular force-constants transformation
The molecular force constants, φii′
αβ (lκ;l′κ′)can be evaluated in terms of the forces on the atoms
(or interaction sites) making up the rigid molecule, using the framework originally developed by
Venkataraman and Sahni,[124] and corrected by others in subsequent work.[212, 213] The necessary
molecular derivatives (both rotational and translational) are expressed as appropriate sums involving
the corresponding derivatives applied to the interaction sites, which can be evaluated analytically.
This approach is far preferable and faster compared to (a not uncommon) one based on numerical
differentiation of the molecules’ positions and orientations. Defining φαβ (lκk;l′κ′k′)as the atomic
force-constant corresponding to the interaction between atoms kand k′on the molecules lκ and l′κ′,
the relevant molecular force-constant equations can be rewritten in a tensor form as follows[124, 212,
213]
φtt(lκ;l′κ′) = X
kk′
φ(lκk;l′κ′k′)(A.1a)
φtr(lκ;l′κ′) = X
kk′
φ(lκk;l′κ′k′)R(k′)(A.1b)
φrt(lκ;l′κ′) = X
kk′RT(k)φ(lκk;l′κ′k′)(A.1c)
φrr(lκ;l′κ′) = X
kk′RT(k)φ(lκk;l′κ′k′)R(k′)
+δll′δκκ′X
k
[X(κk)·F(lκk)I3−X(κk)⊗F(lκk)] (A.1d)
where F(lκk)is the the force on atom kassociated with molecule κin cell l. Also Ris an anti-
symmetric matrix defined as
R(κk) =
0Xz−Xy
−Xz0Xx
Xy−Xx0
(A.2)
where the Xαis the αcoordinate of the position of atom kwith respect to the CM of molecule κ.
The atomic-to-molecular transformations of the force-constants above can be directly used for
interaction between two distinct molecules because it consists of atomic force constants between
distinct atoms (which can be expressed as second derivatives). However, for molecular self terms, the
104
transformation can be divided into self and non-self atomic terms as follows
φtt(lκ;lκ) = X
k
φ(lκk;lκk) + X
k6=k′
φ(lκk;lκk′)(A.3a)
φtr(lκ;lκ) = X
k
φ(lκk;lκk)R(κk)
+X
k6=k′
φ(lκk;lκk′)R(k′κ′)(A.3b)
φrt(lκ;lκ) = X
kRT(κk)φ(lκk;lκk)
+X
k6=k′RT(κk)φ(lκk;lκk′)(A.3c)
φrr(lκ;lκ) = X
kRT(κk)φ(lκk;lκk)R(κk)
+X
k6=k′RT(κk)φ(lκk;lκk′)R(κ′k′)
+X
k
[X(κk)·F(lκk)I3−X(κk)⊗F(lκk)] (A.3d)
where φ(lκk;lκk)is the self atomic force-constant tensor of atom k; which is given by the sum rule
as follows
φ(lκk;lκk) = −X
l′κ′k′
κ′6=κ
φ(lκk;l′κ′k′).(A.4)
105
APPENDIX B
Atomic Hessian matrix of TIP4P water model
The atomic force-constant coefficient between atom kin cell land atom k′in cell l′is defined as[28,
29, 123]
φαβ(lk;l′k′)≡∂2U
∂uα(lk)∂uβ(l′k′)0
(B.1)
where it is assumed that the two atoms are different; i.e. non-self (Eq. (A.4) is used for self-terms).
For clarity, lk and l′k′atoms will be called iand jfor the rest of this section.
The potential energy of the TIP4P water model can be decomposed into ULJ, for Lennard-Jones
O-O interactions, and Ucoul for the Coulomb interactions between the other atoms; then
U=ULJ +Ucoul.(B.2)
Let us now start with the second derivatives of the LJ potential,
uij,m = 4ǫ"σ
rij,m 12
−σ
rij,m 6#(B.3)
which is defined as
ULJ =X
i<j X
m
uij,m +1
2X
iX
m6=0
uii,m (B.4)
where rij,m =ri−rj+Rm, where Rmis m’s supercell coordinate, which is defined in terms of the
supercell lattice vectors, ai, as follows
Rm=m1a1+m2a2+m3a3, mi= 0,1,2, ...
Here, iand jsum over Lennard-Jones sites (oxygen atoms) in the supercell. The force on atom iis
given as a first-derivative of (B.4) as follows
Fi=−X
j=1
j6=iX
m
u′
ij,m
rij,m
rij,m (B.5)
where, u′
ij,m is the first derivative of uij,m. The respective αβ component of the second derivative of
ij atoms pair is given by
φLJ
αβ (i, j) = X
mvij,m −wij,m
r4
ij,m
rij,m,αrij,m,β −vij,m
r2
ij,m
δαβ ,(B.6)
where vij,m ≡rij,mu′
ij,m,wij,m ≡r2
ij,mu′′
ij,m and δαβ is the Kronecker delta function.
The long-range Coulomb interaction is treated using the Ewald summation (ES) method.[134, 1]
The ES potential energy of a molecular system modeled as point charges (e.g. TIP4P) is given by
U=Ureal + [Urec −Uintra]−Uself (B.7)
where Ureal and Urec are the ES in the real and reciprocal spaces, respectively, and Uself is the correction
for the self interactions. Since in the standard ES method the reciprocal summation assumes only point
charges, then an additional term, −Uintra , has to be added to cancel out the intramolecular interactions
considered in Urec. The ES energy contributions are given by
Ureal =X
i<j
i,j6∈same mol. X
m
QiQj
erfc (αrij,m)
rij,m
+1
2X
iX
m6=0
Q2
i
erfc (αrii,m)
rii,m
(B.8a)
Urec =2π
VX
G6=0
exp −(G/2α)2
G2X
j=1
QjeiG·rj
2
(B.8b)
Uself =α
√πX
j=1
Q2
j(B.8c)
Uintra =X
i<j
i,j∈same mol.
QiQj
erf (rij )
rij
(B.8d)
107
where αis an adjustable parameter, Qjis the charge of point j;Gis the supercell wave vector,
G≡ |G|. The forces on atom idue to ES potential are given by
Fi=Freal
i+Frec
i−Fintra
i
where the individual components are the negative of the first derivative of ES contributions (i.e. Eqs.
(B.8))
Freal
i=X
j=1
j6=iX
m
QiQj
r3
ij,m erfc (αrij,m) + 2αrij,m
√πrij,m
Frec
i=4πQi
VX
G6=0
Gexp(−G2/4α2)
G2X
j=1
j6=iX
l
Qjsin (G·rij,l)
Fintra
i=X
j∈same mol.
j6=i
QiQjerf (αrij )
rij −2α
√πexp −α2r2
ijrij
r2
ij
The respective αβ component of the second derivatives of ij atoms pair is, then, given by
φES
αβ(ij) = φreal
αβ (ij) + φrec
αβ(ij)−φintra
αβ (ij)(B.10)
where
φreal
αβ (ij) = X
m
QiQj" erfc (αrij,m )
r3
ij,m
+2α
√π
exp −α2r2
ij,m
r2
ij,m !δαβ
−rij,m,αrij,m,β 6α
√π
exp −α2r2
ij,m
r4
ij,m
+4α3
√π
exp −α2r2
ij,m
r2
ij,m
+3erfc (αrij,m)
r5
ij,m !#,(B.11a)
φrec
αβ (ij) = QiQj
4π
VX
G6=0
GαGβcos (G·rij)exp(−G2/4α2)
G2,(B.11b)
φintra
αβ (ij) = QiQj" erf (αrij )
r3
ij −2α
√π
exp −α2r2
ij
r2
ij !δαβ
+rij,αrij,β 6α
√π
exp −α2r2
ij
r4
ij
+4α3
√π
exp −α2r2
ij
r2
ij −3erf (αrij )
r5
ij !# (B.11c)
It is interesting to notice here that although there are no explicit intramolecular interactions within
rigid molecules, their respective force constants do not vanish when ES is used. This is because the
reciprocal part of ES assumes infinite system of periodic images of the supercells. Therefore, all other
images contribute with finite values to the intramolecular force constants.
108
APPENDIX C
Reduction theorem of the full system Hessian matrix
The only nontrivial task to get the harmonic FE (2.15) is to evaluate the system eigenvalues. How-
ever, diagonalizing the full supercell force-constant matrix, Φ, is inefficient as it does not take the
advantage of the translational invariance and periodicity nature of that matrix. Therefore, the goal is
to reduce the big matrix Φto smaller matrices (each of size 3n×3nas shown below) which share
the same eigenvalues with the big matrix. Fortunately, Φis a circulant matrix and such reduction is
possible. The mathematical origin of this “reduction theorem” of circulant matrices was first intro-
duced by Friedman [214] and then applied to 1Dharmonic system[215]. An extension to 3Dsystem
is presented in this section.
Assume a general Bravais crystal with lattice vectors a1,a2and a3; the periodicity of the crystal
in 3Dis manifested by existence of parallel planes perpendicular to the three lattice vectors. The
three vectors along which these planes are repeated are the reciprocal lattice vectors, b1,b2and b3.
Assume now a super-cell with C1,C2, and C3unit cells in each lattice vector direction. Then, the unit
cell coordinates for any fixed point on each cell (box corner is used here) is given by
R(l1l2l3)≡R1+R2+R3≡l1a1+l2a2+l3a3(C.1)
where l1= 0,1, ...C1−1,l2= 0,1, ...C2−1and l3= 0,1, ...C3−1. In all below, the origin used to
be of the first unit cell; i.e. R(0,0,0) = (0,0,0). The allowed wave vectors in such crystal are
k≡k1+k2+k3≡m1b1+m2b2+m3b3(C.2)
where m1= 0,1, ...C1−1,m2= 0,1, ...C2−1, and m3= 0,1, ...C3−1(or, to be within any unit
cell of the reciprocal lattice; usually, the Brillouin zone (centered around Γpoint) is considered). To
apply the reduction technique for 3Dcrystal, one direction is to be considered at a time. Let us take
b1as the negative of the first direction of periodicity of planes of unit cells; in this case, our new unit
cell is a collection of the original unit cells that lie on these planes. This can be seen as a 1Dlinear
chain system along b1direction. The force constant matrix, Φ, can be rearranged in a block form of
interaction between those planar cells
Φ=Φ(1) =
φ(1)(0; 0) ··· φ(1)(0; C1−1)
φ(1)(1; 0) ··· φ(1)(1; C1−1)
.
.
.....
.
.
φ(1)(C1−1; 0) ··· φ(1) (C1−1; C1−1)
(C.3)
where φ(1)(l1;l′
1)represents the force constant matrix (of size 3nC2C3×3nC2C3) between planer
cells l1and l′
1. Two important features of Φmakes it a circulant; first, the force-constants across
different unit-cells is translational invariance, i.e. φ(1)(l;l′) = φ(1)(l′−1), the second property is the
periodic-boundary condition of the supercell, i.e. φ(1)(0; C1−1) = φ(1)(1; 0). Since φ(1) (l1;l′
1)is a
circulant matrix, it can be commuted with any other circulant matrix (i.e. ΦT=TΦ); therefore, both
matrices share the same eigenvectors. If we choose the other circulant matrix to be the permutation
matrix, then the block eigenvectors, v(1)(k, s), for each kis
v(1)(k, s) =
u(1)(k, s)eik·R1(0)
u(1)(k, s)eik·R1(1)
.
.
.
u(1)(k, s)eik·R1(C1−1)
(C.4)
where s= 1,2, ..., 3nis the mode (or branch) index (acoustic, optic, longitudinal and transverse) and
u(1)(k, s)is the displacement vector (of size 3nC2C3×1) of the planar unit cells (independent on
110
which planar unit cell is considered) and R1(l1) = l1a1is the coordinate of lth
1planar unit cell. Note
that k·R1(l1) = k1·R1(l1) = 2πl1m1. Plugging this result into the eigenvalue problem
Φ(1)v(1)(k, s) = λ(k, s)v(1)(k, s)(C.5)
we get
C1−1
X
l′
1=0
φ(1)(l1;l′
1)eik·R1(l′
1)
u(1)(k, s) =
λ(k, s)eik·R1(l1)u(1)(k, s)(C.6)
for each l1; but since this summation is a function of only the index difference, then the summation
is independent on the choice of l1; therefore this equation can be written in the following eigenvalue
problem form
φ(1)
R(k)u(1)(k, s) = λ(k, s)u(1)(k, s)(C.7)
where φ(1)
R(k)is the first reduced matrix (of same size as φ(1) (l1;l′
1)) defined as
φ(1)
R(k) =
C1
X
l′
1=1
φ(1)(l1;l′
1)eik·R1(l′
1)(C.8)
Next, to reduce the problem one more time, the matrix φ(1) (l1;l′
1)can be written in a reduced block
form of force constant matrices of interaction between line of cells (within a plane perpendicular to
b1) perpendicular to b2as following
φ(1)(l1;l′
1) =
φ(2)(l11; l′
11) ··· φ(2)(l11; l1C2)
φ(2)(l12; l′
11) ··· φ(2)(l12; l′
1C2)
.
.
.....
.
.
φ(2)(l1C2;l′
11) ··· φ(2)(l1C2;l′
1C2)
(C.9)
where φ(2)(l1l2;l′
2l′
2)represents the force constant matrix (of size 3nC3×3nC3) between linear cells
l2and l′
2. Since φ(1) (l1;l′
1)is a circulant matrix, then φR(k)is also circulant and the reduction can be
111
applied to Eq. C.8 to get this form of u(1)(k, s)
u(1)(k, s) =
u(2)(k, s)eik·R2(1)
u(2)(k, s)eik·R2(2)
.
.
.
u(2)(k, s)eik·R2(C2)
(C.10)
where u(2)(k, s)is the displacement vector (of size 3nC3×1) of the linear unit cells (again, inde-
pendent on which linear cell is considered). Plugging this result into the eigenvalue problem, Eq.
C.6
φ(2)
R(k)u(2)(k, s) = λ(k, s)u(2) (k, s)(C.11)
where φ(2)
R(k)is the second reduced matrix and defined as
φ(2)
R(k) = X
l′
1l′
2
φ(l1l2;l′
1l′
2)eik·(R1(l′
1)+R2(l′
2))(C.12)
Lastly, the problem can be reduced one more time in b3direction to obtain the smallest reduced form
possible. The resultant formula of u(2)(k, s)is
u(2)(k, s) =
u(3)(k, s)eik·R3(1)
u(3)(k, s)eik·R3(2)
.
.
.
u(3)(k, s)eik·R3(C3)
(C.13)
where u(3)(k, s) = e(k, s)is the displacement (or polarization) vector (of size 3n×1). Using this
form of u(2)(k, s), Eq. C.11 can be reduced to
φ(3)
R(k)e(k, s) = λ(k, s)e(k, s)(C.14)
where, φ(3)
R(k) (will simply be called φ(3)
R(k)from now on) is the last reduced reduced matrix form in
3D(of size 3n×3n) and defined as
φR(k)≡φ(3)
R(k) = X
l′
1l′
2l′
3
φ(l1l2l3;l′
1l′
2l′
3)eik·R(l′
1l′
2l′
3)(C.15)
112
where φ(l1l2l3;l′
1l′
2l′
3)is the original cell-cell force constant matrix between cells of indices l1l2l3and
l′
1l′
2l′
3with size same size as φR(k). Therefore, the problem of diagonalizing the original 3nC1C2C3×
3nC1C2C3,Φ, (Eq. C.5) is reduced to diagonalizing C1C2C3matrices, φR(k), each has size of only
3n×3n; the eigenvalues of these small matrices are exactly the same of the original big matrix, Φ.
We have shown above that the force-constants matrix, Φ, is symmetric; an important property of
φR(k)that is closely related to symmetry is Hermiticity; which allows using special/efficient routines
of diagonalization that take advantage of such property. The problem can be simplified even more by
noticing the inversion symmetry property of the reduced matrix[29]
φR(−k) = φ∗
R(k) = φR(k)(C.16)
which means that both φR(k)and φR(−k)have the same eigenvalues (degeneracy). Therefore, only
subset of the reduced matrices has to be diagonalized.
113
APPENDIX D
Equivalence of different FE formulas
The equation of motion of a molecular system, in the Fourier space, with respect to a fixed frame of
reference, is given by [124]
Φu=ω2Mu (D.1)
where Mis a block diagonal matrix given by
M=
m1
I1
...
mN
IN
6N×6N
(D.2)
where miis a 3×3diagonal tensor of ith molecule mass and Iiis a 3×3tensor of molecule ith
moment of inertia. Since Mis not, in general, diagonal (because Iiare full tensors) it is not possible
to write Eq. D.1 as a standard eigenvalue problem (i.e. Ax =λx). Fortunately, changing the frame of
reference to a body-fixed one, with it’s axes parallel to the principal axes of the respective molecule
modifies the equation of motion to a symmetric form that has the same frequencies, ω, as the original
problem[124]. Applying such transformation to Eq. D.1 yields
ΦPuP=ω2MPuP
where,
ΦP=ATΦA,MP=ATMA,uP=ATu
MPis now a diagonal tensor with the same form as Mexcept that Ii=IPis diagonal tensor of prin-
cipal moments of inertia (independent on which molecule is considered), ΦPis the second-derivatives
matrix taken with respect to the new principal axes, uPis the deviation vector (both linear and angu-
lar) taken from the new frame of reference and Ais an orthogonal matrix (represents the eigenvectors
matrix of M) of transformation from the space-fixed to the body-fixed frame. It can also be written
in the regular eigenvalue problem as follows
˜
ΦP˜
uP=ω2˜
uP
where,
˜
ΦP=1
√MP
ΦP
1
√MP
,and ˜
uP=√MPuP
To show equivalence between our method (using λi) and the dynamical matrix approach (using
ωi), let us write Uharm in terms of ˜
ΦPas follows
Uharm =1
2
6N
X
i=1
˜
xT
P˜
ΦP˜
xP(D.3)
where ˜
xis the inertia-weighted molecular displacements vector (both linear and angular) measured
in the new frame and given by
˜
xP=√MPATx
and xis the respective displacements in the space-fixed frame with the rotational part defined through
the new variables κi.
In the CM system, the “momentum” contribution to the partition function is given by
PCM =2πmkBT
h23(N−1)
2"(2πkBT)3/2VΩ(I1I2I3)1/2
h3#N
(D.4)
115
and the “configurational” partition function, Z, is given by
Z=1
VN
ΩZexp "−β1
2
6N
X
i=1
xTΦx#dx(D.5)
or, in the new frame,
Z=1
p|MP|
1
VN
ΩZexp "−β1
2
6N
X
i=1
˜
xT
P˜
ΦP˜
xP#d˜
xP(D.6)
where
d˜
xP=p|MP|ATdx,AT= 1
and
˜
MP=
3N−3
Y
i=1
mi×
3N
Y
i=1
Ii
This integration can be easily evaluated by diagonalizing ˜
ΦP; then, Z(fixing the CM) is given in
terms of the eigenvalues, ωi, and eigenvectors, ηi, of ˜
ΦPby
ZCM =1
r
˜
MP
1
VN
ΩZexp "−β1
2
6N−3
X
i=1
ω2
iη2
i#dη1...dηN−1(D.7)
which can be easily evaluated to
ZCM =1
r
˜
MP
1
VN
Ω
6N−3
Y
i=1 s2π
βω2
i
(D.8)
Thus, using Eqs. D.4 and D.8, the total partition function in the CM system is
QCM =PC M ZCM
="2πkBT
h23/2#N−1"(2πkBT)3/2
sh3#N
×
6N−3
Y
i=1 s2π
βω2
i
=
6(N−1)
Y
i=1 1
β~ωi(D.9)
Then, the respective free energy (beyond Srot and Sdis) is
ACM
harm =kBT
6(N−1)
X
i=1
ln ~ωi
kBT(D.10)
116
which is, again, the common formula for harmonic free energy; that is, both representations of the
free energy ( in terms of λiand ωi) for free energy are actually equivalent. However, if the whole
purpose of calculations is to estimate the free energy, then the formula that involves λiis much easier
to calculate since it does not involve considering the inertia matrix, M, at all.
117
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